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abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.13375">arXiv:2411.13375</a> <span> [<a href="https://arxiv.org/pdf/2411.13375">pdf</a>, <a href="https://arxiv.org/format/2411.13375">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The weight hierarchy of decreasing norm-trace codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.13375v1-abstract-short" style="display: inline;"> The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have bee… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13375v1-abstract-full').style.display = 'inline'; document.getElementById('2411.13375v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13375v1-abstract-full" style="display: none;"> The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$ over the finite field of cardinality $q^s$, where $u$ is a positive divisor of $\frac{q^s - 1}{q - 1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13375v1-abstract-full').style.display = 'none'; document.getElementById('2411.13375v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05; 11T71; 14G50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.02916">arXiv:2408.02916</a> <span> [<a href="https://arxiv.org/pdf/2408.02916">pdf</a>, <a href="https://arxiv.org/ps/2408.02916">ps</a>, <a href="https://arxiv.org/format/2408.02916">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Binary Triorthogonal and CSS-T Codes for Quantum Error Correction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a>, <a href="/search/cs?searchtype=author&query=Soprunov%2C+I">Ivan Soprunov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.02916v1-abstract-short" style="display: inline;"> In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a cons… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02916v1-abstract-full').style.display = 'inline'; document.getElementById('2408.02916v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02916v1-abstract-full" style="display: none;"> In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a consequence, we show that a binary triorthogonal matrix uniquely determines the parameters of the corresponding triorthogonal quantum code, meaning that any other equivalent matrix that is also triorthogonal gives rise to a triorthogonal quantum code with the same parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02916v1-abstract-full').style.display = 'none'; document.getElementById('2408.02916v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.11810">arXiv:2407.11810</a> <span> [<a href="https://arxiv.org/pdf/2407.11810">pdf</a>, <a href="https://arxiv.org/ps/2407.11810">ps</a>, <a href="https://arxiv.org/format/2407.11810">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> About the generalized Hamming weights of matrix-product codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.11810v1-abstract-short" style="display: inline;"> We derive a general lower bound for the generalized Hamming weights of nested matrix-product codes, with a particular emphasis on the cases with two and three constituent codes. We also provide an upper bound which is reminiscent of the bounds used for the minimum distance of matrix-product codes. When the constituent codes are two Reed-Solomon codes, we obtain an explicit formula for the generali… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11810v1-abstract-full').style.display = 'inline'; document.getElementById('2407.11810v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.11810v1-abstract-full" style="display: none;"> We derive a general lower bound for the generalized Hamming weights of nested matrix-product codes, with a particular emphasis on the cases with two and three constituent codes. We also provide an upper bound which is reminiscent of the bounds used for the minimum distance of matrix-product codes. When the constituent codes are two Reed-Solomon codes, we obtain an explicit formula for the generalized Hamming weights of the resulting matrix-product code. We also deal with the non-nested case for the case of two constituent codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11810v1-abstract-full').style.display = 'none'; document.getElementById('2407.11810v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05 (Primary) 94B65; 11T71 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.17518">arXiv:2312.17518</a> <span> [<a href="https://arxiv.org/pdf/2312.17518">pdf</a>, <a href="https://arxiv.org/ps/2312.17518">ps</a>, <a href="https://arxiv.org/format/2312.17518">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11128-024-04427-5">10.1007/s11128-024-04427-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Camps-Moreno%2C+E">Eduardo Camps-Moreno</a>, <a href="/search/cs?searchtype=author&query=L%C3%B3pez%2C+H+H">Hiram H. L贸pez</a>, <a href="/search/cs?searchtype=author&query=Matthews%2C+G+L">Gretchen L. Matthews</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a>, <a href="/search/cs?searchtype=author&query=Soprunov%2C+I">Ivan Soprunov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.17518v2-abstract-short" style="display: inline;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17518v2-abstract-full').style.display = 'inline'; document.getElementById('2312.17518v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.17518v2-abstract-full" style="display: none;"> CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $(C_1, C_2)$ is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed-Muller, cyclic, and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17518v2-abstract-full').style.display = 'none'; document.getElementById('2312.17518v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05 (Primary); 81P70; 11T71; 14G50 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Quantum Inf Process 23, 230 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.15308">arXiv:2312.15308</a> <span> [<a href="https://arxiv.org/pdf/2312.15308">pdf</a>, <a href="https://arxiv.org/ps/2312.15308">ps</a>, <a href="https://arxiv.org/format/2312.15308">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.15308v3-abstract-short" style="display: inline;"> Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.15308v3-abstract-full').style.display = 'inline'; document.getElementById('2312.15308v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.15308v3-abstract-full" style="display: none;"> Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.15308v3-abstract-full').style.display = 'none'; document.getElementById('2312.15308v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81P70 (Primary) 94B05; 13P25 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13921">arXiv:2312.13921</a> <span> [<a href="https://arxiv.org/pdf/2312.13921">pdf</a>, <a href="https://arxiv.org/ps/2312.13921">ps</a>, <a href="https://arxiv.org/format/2312.13921">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> Hulls of projective Reed-Muller codes over the projective plane </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13921v3-abstract-short" style="display: inline;"> By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed-Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13921v3-abstract-full').style.display = 'inline'; document.getElementById('2312.13921v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13921v3-abstract-full" style="display: none;"> By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed-Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have all the parameters of the symmetric and asymmetric entanglement-assisted quantum error-correcting codes constructed with projective Reed-Muller codes over the projective plane. As a byproduct, we also compute the dimension of the Hermitian hull for affine Reed-Muller codes in 2 variables. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13921v3-abstract-full').style.display = 'none'; document.getElementById('2312.13921v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81P70 (Primary) 94B05; 13P25 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.05072">arXiv:2312.05072</a> <span> [<a href="https://arxiv.org/pdf/2312.05072">pdf</a>, <a href="https://arxiv.org/ps/2312.05072">ps</a>, <a href="https://arxiv.org/format/2312.05072">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1109/tit.2024.3483998">10.1109/tit.2024.3483998 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A recursive construction for projective Reed-Muller codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.05072v2-abstract-short" style="display: inline;"> We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.05072v2-abstract-full').style.display = 'inline'; document.getElementById('2312.05072v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.05072v2-abstract-full" style="display: none;"> We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.05072v2-abstract-full').style.display = 'none'; document.getElementById('2312.05072v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05 (Primary) 11T71; 14G50 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.09298">arXiv:2307.09298</a> <span> [<a href="https://arxiv.org/pdf/2307.09298">pdf</a>, <a href="https://arxiv.org/ps/2307.09298">ps</a>, <a href="https://arxiv.org/format/2307.09298">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.ffa.2023.102353">10.1016/j.ffa.2023.102353 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Subfield subcodes of projective Reed-Muller codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gimenez%2C+P">Philippe Gimenez</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.09298v2-abstract-short" style="display: inline;"> Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gr枚bner basis for the vanishing ideal of the set of standard repr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.09298v2-abstract-full').style.display = 'inline'; document.getElementById('2307.09298v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.09298v2-abstract-full" style="display: none;"> Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gr枚bner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gr枚bner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.09298v2-abstract-full').style.display = 'none'; document.getElementById('2307.09298v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11T71 (Primary) 94B05; 14G50; 13P25 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.08121">arXiv:2304.08121</a> <span> [<a href="https://arxiv.org/pdf/2304.08121">pdf</a>, <a href="https://arxiv.org/ps/2304.08121">ps</a>, <a href="https://arxiv.org/format/2304.08121">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s40314-023-02506-4">10.1007/s40314-023-02506-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gimenez%2C+P">Philippe Gimenez</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2304.08121v2-abstract-short" style="display: inline;"> We study the subfield subcodes of projective Reed-Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.08121v2-abstract-full" style="display: none;"> We study the subfield subcodes of projective Reed-Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.08121v2-abstract-full').style.display = 'none'; document.getElementById('2304.08121v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81P70 (Primary) 94B05; 13P25 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.04683">arXiv:2202.04683</a> <span> [<a href="https://arxiv.org/pdf/2202.04683">pdf</a>, <a href="https://arxiv.org/ps/2202.04683">ps</a>, <a href="https://arxiv.org/format/2202.04683">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s40863-022-00330-y">10.1007/s40863-022-00330-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Saturation and vanishing ideals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gimenez%2C+P">Philippe Gimenez</a>, <a href="/search/cs?searchtype=author&query=Ruano%2C+D">Diego Ruano</a>, <a href="/search/cs?searchtype=author&query=San-Jos%C3%A9%2C+R">Rodrigo San-Jos茅</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.04683v2-abstract-short" style="display: inline;"> We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space $\mathbb{P}^{m-1}$. We concern ourselves with the problem of computing the vanishing ideal $I(\mathbb{X})$. This is usually done by adding the equations of the projective space… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.04683v2-abstract-full').style.display = 'inline'; document.getElementById('2202.04683v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.04683v2-abstract-full" style="display: none;"> We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space $\mathbb{P}^{m-1}$. We concern ourselves with the problem of computing the vanishing ideal $I(\mathbb{X})$. This is usually done by adding the equations of the projective space $I(\mathbb{P}^{m-1})$ to $I$ and computing the radical. We give an alternative and more efficient way using the saturation with respect to the homogeneous maximal ideal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.04683v2-abstract-full').style.display = 'none'; document.getElementById('2202.04683v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div 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