%!PS-Adobe-2.0 %%Creator: dvips(k) 2023.1 (TeX Live 2023) Copyright 2023 Radical Eye Software %%Title: arXiv:2412.20771v1 [cs.IT] 30 Dec 2024 %%CreationDate: Tue Dec 31 05:31:16 2024 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z main.dvi -o %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2024.12.31:0531 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: l3backend-dvips.pro 0 0 %% %% This is file `l3backend-dvips.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% l3backend-header.dtx (with options: `header,dvips') %% %% Copyright (C) 2019-2023 The LaTeX Project %% %% It may be distributed and/or modified under the conditions of %% the LaTeX Project Public License (LPPL), either version 1.3c of %% this license or (at your option) any later version. 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0 TeXcolorgray 1237 2104 a SDict begin H.R end 1237 2104 a 1237 2163 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.gao2010additive) cvn H.B /ANN pdfmark end 1237 2163 a FC(,)1291 2163 y SDict begin H.S end 1291 2163 a 0 TeXcolorgray FC(77)p 0 TeXcolorgray 1382 2104 a SDict begin H.R end 1382 2104 a 1382 2163 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.wu2012reduced) cvn H.B /ANN pdfmark end 1382 2163 a FC(,)1435 2163 y SDict begin H.S end 1435 2163 a 0 TeXcolorgray FC(43)p 0 TeXcolorgray 1526 2104 a SDict begin H.R end 1526 2104 a 1526 2163 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.lin2016fft) cvn H.B /ANN pdfmark end 1526 2163 a FC(,)1580 2163 y SDict begin H.S end 1580 2163 a 0 TeXcolorgray FC(21)p 0 TeXcolorgray 1671 2104 a SDict begin H.R end 1671 2104 a 1671 2163 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype 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FC(within)27 b(the)j(Hamming)f(distance)g FB(\032n)g FC(of)h Fp(y)s FC(.)40 b(Due)30 b(to)g(Johnson-b)s(ound)d([)3147 2502 y SDict begin H.S end 3147 2502 a 0 TeXcolorgray FC(37)p 0 TeXcolorgray 3238 2443 a SDict begin H.R end 3238 2443 a 3238 2502 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.johnson1962new) cvn H.B /ANN pdfmark end 3238 2502 a FC(],)j(Reed{Solomon)0 2614 y(co)s(des)g(are)h(\()p FB(\032;)15 b(L)p FC(\)-list-deco)s(dable) 30 b(for)g(error)g(parameter)h FB(\032)25 b FC(=)g(1)c FA(\000)2343 2538 y(p)p 2419 2538 70 4 v 76 x FB(R)g FA(\000)f FB(\017)30 b FC(and)g(list)f(size)h FB(L)25 b FC(=)g FA(O)3441 2541 y Fn(\000)3493 2579 y Fx(1)p 3493 2594 36 4 v 3496 2646 a Fy(\017)3538 2541 y Fn(\001)3579 2614 y FC(.)141 2739 y(Sudan[)415 2739 y SDict begin H.S end 415 2739 a 0 TeXcolorgray FC(66)p 0 TeXcolorgray 505 2680 a SDict begin H.R end 505 2680 a 505 2739 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.sudan1997decoding) cvn H.B /ANN pdfmark end 505 2739 a FC(])e(dev)m(elop)s(ed)f(an)h(e\016cien)m(t)g (algorithm)f(to)h(list-deco)s(de)f(RS)g(co)s(des)h(up)f(to)h(the)g (radius)e FB(\032)26 b FC(=)f(1)5 b FA(\000)3684 2662 y(p)p 3760 2662 116 4 v 77 x FC(2)p FB(R)q FC(,)0 2852 y(whic)m(h)22 b(w)m(as)i(later)f(impro)m(v)m(ed)f(b)m(y)h(Sudan)f(and)h (Gurusw)m(ami)e([)2076 2852 y SDict begin H.S end 2076 2852 a 0 TeXcolorgray FC(28)p 0 TeXcolorgray 2167 2793 a SDict begin H.R end 2167 2793 a 2167 2852 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.guruswami1999improved) cvn H.B /ANN pdfmark end 2167 2852 a FC(])j(to)g(e\016cien)m(tly)e(list-deco)s(de)h(RS)f(co)s (des)i(up)e(to)i(the)0 2965 y(Johnson)j(radius)f(1)17 b FA(\000)772 2888 y(p)p 848 2888 70 4 v 77 x FB(R)q FC(.)40 b(Ko)s(etter)29 b(and)e(V)-8 b(ardy)29 b([)1785 2965 y SDict begin H.S end 1785 2965 a 0 TeXcolorgray FC(39)p 0 TeXcolorgray 1876 2906 a SDict begin H.R end 1876 2906 a 1876 2965 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.koetter2003algebraic) cvn H.B /ANN pdfmark end 1876 2965 a FC(])f(further)f(mo)s (di\014ed)f(the)j(Gurusw)m(ami-Sudan)c(algorithm)0 3078 y(to)31 b(incorp)s(orate)f(soft-deco)s(ding.)141 3190 y(F)-8 b(or)41 b Fp(v)j FA(2)d Fz(F)561 3158 y Fy(n)561 3213 y(q)614 3190 y FC(,)i(let)d FB(B)892 3204 y Fy(t)921 3190 y FC(\()p Fp(v)s FC(\))h(b)s(e)e(the)h(radius-)p FB(t)f FC(ball)f(cen)m(tered)j(at)g Fp(v)s FC(.)69 b(A)40 b(co)s(de)g FA(C)45 b FC(is)39 b(\()p FB(\032n;)15 b(N)10 b FC(\)-sequence)42 b(re-)0 3303 y(constructable)35 b([)591 3303 y SDict begin H.S end 591 3303 a 0 TeXcolorgray FC(42)p 0 TeXcolorgray 682 3245 a SDict begin H.R end 682 3245 a 682 3303 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.levenshtein2001efficientA) cvn H.B /ANN pdfmark end 682 3303 a FC(,)743 3303 y SDict begin H.S end 743 3303 a 0 TeXcolorgray FC(41)p 0 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Fy(\032n)2855 3538 y FC(\()p Fp(y)2944 3552 y Fy(i)2973 3538 y FC(\).)141 3651 y(Recen)m(tly)-8 b(,)36 b(Singh)m(vi)c(et)j(al.) 51 b([)1147 3651 y SDict begin H.S end 1147 3651 a 0 TeXcolorgray FC(62)p 0 TeXcolorgray 1239 3593 a SDict begin H.R end 1239 3593 a 1239 3651 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.singhvi2024optimal) cvn H.B /ANN pdfmark end 1239 3651 a FC(])34 b(explored)f(RS)g(co)s(des)h(in)f(the)h(con)m(text)i(of) e(sequence)h(reconstruction)f(and)0 3764 y(prop)s(osed)29 b(an)h(e\016cien)m(t)h(algorithm)f(capable)g(of)g(deco)s(ding)g(b)s(ey) m(ond)f(the)i(Johnson)e(radius.)0 3909 y SDict begin H.S end 0 3909 a 0 3909 a SDict begin 13.6 H.A end 0 3909 a 0 3909 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.2) cvn /DEST pdfmark end 0 3909 a 99 x Fo(2.2)112 b(RS)38 b(co)s(des)f(against)h(ins-del)f(errors)0 4179 y FC(The)f(p)s (erformance)f(of)h(RS)f(co)s(des)i(against)f(ins-del)e(errors)h(w)m(as) i(\014rst)e(considered)g(in)g([)3072 4179 y SDict begin H.S end 3072 4179 a 0 TeXcolorgray FC(56)p 0 TeXcolorgray 3163 4121 a SDict begin H.R end 3163 4121 a 3163 4179 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.safavi2002traitor) cvn H.B /ANN pdfmark end 3163 4179 a FC(])h(in)f(the)h(con)m(text)i(of)0 4292 y(traitor)k(tracing.)74 b(In)40 b([)826 4292 y SDict begin H.S end 826 4292 a 0 TeXcolorgray FC(75)p 0 TeXcolorgray 918 4234 a SDict begin H.R end 918 4234 a 918 4292 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.wang2004deletion) cvn H.B /ANN pdfmark end 918 4292 a FC(],)k(the)e(authors)f(constructed) h(a)g([5)p FB(;)15 b FC(2]-RS)43 b(co)s(de)e(capable)h(of)g(correcting) f(a)h(single)0 4405 y(deletion.)56 b(Subsequen)m(tly)-8 b(,)36 b(in)e([)1109 4405 y SDict begin H.S end 1109 4405 a 0 TeXcolorgray FC(71)p 0 TeXcolorgray 1200 4347 a SDict begin H.R end 1200 4347 a 1200 4405 a SDict begin 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FB(X)1290 1288 y Fy(I)1321 1297 y Fj(2)1498 1274 y FB(:)15 b(:)g(:)105 b(X)1790 1236 y Fy(k)r Fv(\000)p Fx(1)1783 1303 y Fy(I)1814 1312 y Fj(2)2084 1274 y FB(X)2159 1288 y Fy(J)2198 1297 y Fj(2)2376 1274 y FB(:)15 b(:)g(:)109 b(X)2672 1236 y Fy(k)r Fv(\000)p Fx(1)2665 1303 y Fy(J)2704 1312 y Fj(2)1040 1371 y FC(.)1040 1405 y(.)1040 1438 y(.)1274 1371 y(.)1274 1405 y(.)1274 1438 y(.)199 b FB(:)15 b(:)g(:)1803 1371 y FC(.)1803 1405 y(.)1803 1438 y(.)2147 1371 y(.)2147 1405 y(.)2147 1438 y(.)204 b FB(:)15 b(:)g(:)2685 1371 y FC(.)2685 1405 y(.)2685 1438 y(.)1030 1556 y(1)83 b FB(X)1233 1570 y Fy(I)1264 1582 y Fj(2)p Fw(k)q Fi(\000)p Fj(1)1498 1556 y FB(:)15 b(:)g(:)85 b(X)1770 1518 y Fy(k)r Fv(\000)p Fx(1)1763 1585 y Fy(I)1794 1597 y Fj(2)p Fw(k)q Fi(\000)p Fj(1)2028 1556 y FB(X)2103 1570 y Fy(J)2142 1582 y Fj(2)p Fw(k)q Fi(\000)p Fj(1)2376 1556 y FB(:)15 b(:)g(:)84 b(X)2647 1518 y Fy(k)r Fv(\000)p Fx(1)2640 1585 y Fy(J)2679 1597 y Fj(2)p Fw(k)q Fi(\000)p Fj(1)2830 1066 y Fn(1)2830 1226 y(C)2830 1281 y(C)2830 1335 y(C)2830 1390 y(C)2830 1448 y(A)2924 1358 y FB(:)835 b FC(\(1\))0 1663 y SDict begin H.S end 0 1663 a 0 1663 a SDict begin 13.6 H.A end 0 1663 a 0 1663 a SDict begin [/View [/XYZ H.V]/Dest (proposition.1) cvn /DEST pdfmark end 0 1663 a 0 TeXcolorgray 113 x Fq(Prop)s(osition)39 b(1.)p 0 TeXcolorgray 44 w Fr([)721 1776 y SDict begin H.S end 721 1776 a 0 TeXcolorgray Fr(16)p 0 TeXcolorgray 814 1717 a SDict begin H.R end 814 1717 a 814 1776 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.con2023reed) cvn H.B /ANN pdfmark end 814 1776 a Fr(,)c(Pr)-5 b(op)g(osition)38 b(2.1])e(L)-5 b(et)35 b FA(C)40 b Fr(b)-5 b(e)35 b(an)h FC([)p FB(n;)15 b(k)s FC(])2236 1790 y Fy(q)2310 1776 y Fr(RS-c)-5 b(o)g(de)36 b(de\014ne)-5 b(d)36 b(by)f(an)h(evaluation)g(ve)-5 b(ctor)0 1889 y Fp(\013)35 b FC(=)g(\()p FB(\013)303 1903 y Fx(1)343 1889 y FB(;)15 b(:)g(:)g(:)i(;)e(\013)603 1903 y Fy(n)650 1889 y FC(\))p Fr(.)59 b(If)38 b(for)g(every)g(two)i(incr)-5 b(e)g(asing)39 b(ve)-5 b(ctors)39 b FB(I)7 b(;)15 b(J)44 b FA(2)35 b FC([)p FB(n)p FC(])2572 1856 y Fx(2)p Fy(k)r Fv(\000)p Fx(1)2779 1889 y Fr(that)k(agr)-5 b(e)g(e)39 b(on)f(at)h(most)g FB(k)28 b FA(\000)c FC(1)0 2002 y Fr(c)-5 b(o)g(or)g(dinates,)46 b(it)41 b(holds)h(that)g FC(det\()p FB(V)1258 2016 y Fy(I)5 b(;J)1363 2002 y FC(\()p Fp(\013)p FC(\)\))41 b FA(6)p FC(=)f(0)p Fr(,)j(then)f FB(d)2063 2016 y Fy(F)2121 2002 y FC(\()p FA(C)5 b FC(\))41 b(=)f FB(n)26 b FA(\000)g FC(2)p FB(k)j FC(+)d(2)p Fr(,)43 b(i.e.,)f FA(C)k Fr(achieves)41 b(the)g(half-)0 2115 y(Singleton)33 b(b)-5 b(ound.)141 2302 y FC(F)d(or)45 b(the)e(ease)i(of)f(exp)s(osition,)i(w)m(e)e(sa)m(y)g(that)g(an)g([)p FB(n;)15 b(k)s FC(])2149 2316 y Fy(q)2231 2302 y FC(RS-co)s(de)44 b(de\014ned)e(b)m(y)i(an)f(ev)-5 b(aluation)43 b(v)m(ector)0 2415 y Fp(\013)28 b FC(=)g(\()p FB(\013)289 2429 y Fx(1)328 2415 y FB(;)15 b(:)g(:)g(:)i(;)e(\013)588 2429 y Fy(n)636 2415 y FC(\))32 b(satis\014es)f(the)h Fr(algebr)-5 b(aic)35 b(c)-5 b(ondition)41 b FC(if)31 b(the)h(ev)-5 b(aluation)31 b(v)m(ector)j Fp(\013)e FC(satis\014es)f(the)h(condition)0 2528 y(stated)f(in)e(Prop)s(osition)869 2528 y SDict begin H.S end 869 2528 a 0 TeXcolorgray FC(1)p 0 TeXcolorgray 914 2470 a SDict begin H.R end 914 2470 a 914 2528 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (proposition.1) cvn H.B /ANN pdfmark end 914 2528 a FC(.)0 2603 y SDict begin H.S end 0 2603 a 0 2603 a SDict begin 13.6 H.A end 0 2603 a 0 2603 a SDict begin [/View [/XYZ H.V]/Dest (lemma.2) cvn /DEST pdfmark end 0 2603 a 0 TeXcolorgray 113 x Fq(Lemma)47 b(2.)p 0 TeXcolorgray 49 w Fr(L)-5 b(et)44 b FA(C)49 b Fr(b)-5 b(e)44 b(an)g FC([)p FB(n;)15 b FC(2])1234 2730 y Fy(q)1273 2716 y Fr(-RS)44 b(c)-5 b(o)g(de)45 b(with)g(evaluation)g(ve)-5 b(ctor)45 b Fp(\013)h FC(=)g(\()p FB(\013)2940 2730 y Fx(1)2979 2716 y FB(;)15 b(:)g(:)g(:)i(;)e(\013) 3239 2730 y Fy(n)3287 2716 y FC(\))44 b Fr(satisfying)h(the)0 2829 y(algebr)-5 b(aic)34 b(c)-5 b(ondition.)43 b(L)-5 b(et)33 b FB(g)c FC(:)1091 2755 y Fn(\000)1133 2788 y Fl(\013)1142 2860 y Fx(3)1187 2755 y Fn(\001)1254 2829 y FA(!)c Fz(F)1420 2843 y Fy(q)1496 2829 y Fr(b)-5 b(e)33 b(a)g(mapping)h(that)g(assigns)f(the)g(triplet)g FC(\()p FB(\013)3065 2843 y Fy(I)3096 2852 y Fj(1)3135 2829 y FB(;)15 b(\013)3233 2843 y Fy(I)3264 2852 y Fj(2)3303 2829 y FB(;)g(\013)3401 2843 y Fy(I)3432 2852 y Fj(3)3471 2829 y FC(\))33 b Fr(to)1754 3013 y FB(\013)1812 3027 y Fy(I)1843 3036 y Fj(1)1902 3013 y FA(\000)20 b FB(\013)2051 3027 y Fy(I)2082 3036 y Fj(2)p 1754 3054 367 4 v 1754 3137 a FB(\013)1812 3151 y Fy(I)1843 3160 y Fj(2)1902 3137 y FA(\000)g FB(\013)2051 3151 y Fy(I)2082 3160 y Fj(3)2130 3075 y FB(;)0 3337 y Fr(wher)-5 b(e)34 b FB(I)e FA(2)25 b FC([)p FB(n)p FC(])520 3304 y Fx(3)592 3337 y Fr(is)33 b(an)g(incr)-5 b(e)g(asing)33 b(ve)-5 b(ctor.)43 b(Then,)33 b(the)g(mapping)h FB(g)i Fr(is)c(one-to-one)h(over)g(the)g (domain)3644 3264 y Fn(\000)3685 3297 y Fl(\013)3695 3369 y Fx(3)3739 3264 y Fn(\001)3781 3337 y Fr(.)p 0 TeXcolorgray 0 3525 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 47 w FC(Since)42 b FA(C)47 b FC(satis\014es)c(the)f(algebraic)h (condition,)i(it)d(follo)m(ws)g(from)g(Prop)s(osition)3126 3525 y SDict begin H.S end 3126 3525 a 0 TeXcolorgray FC(1)p 0 TeXcolorgray 3171 3466 a SDict begin H.R end 3171 3466 a 3171 3525 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (proposition.1) cvn H.B /ANN pdfmark end 3171 3525 a 43 w FC(that)h(for)g(an)m(y)g(t)m(w)m(o)0 3638 y(increasing)29 b(v)m(ectors)j FB(I)7 b(;)15 b(J)35 b FA(2)25 b FC([)p FB(n)p FC(])1097 3605 y Fx(3)1166 3638 y FC(that)31 b(agree)h(on)e(at)h (most)g(one)g(co)s(ordinate,)f(the)h(follo)m(wing)e(holds:)1494 3766 y Fn(\014)1494 3820 y(\014)1494 3875 y(\014)1494 3930 y(\014)1494 3984 y(\014)1494 4039 y(\014)1525 3770 y(0)1525 3933 y(@)1604 3839 y FC(1)84 b FB(\013)1791 3853 y Fy(I)1822 3862 y Fj(1)1943 3839 y FB(\013)2001 3853 y Fy(J)2040 3862 y Fj(1)1604 3952 y FC(1)g FB(\013)1791 3966 y Fy(I)1822 3975 y Fj(2)1943 3952 y FB(\013)2001 3966 y Fy(J)2040 3975 y Fj(2)1604 4065 y FC(1)g FB(\013)1791 4079 y Fy(I)1822 4088 y Fj(3)1943 4065 y FB(\013)2001 4079 y Fy(J)2040 4088 y Fj(3)2079 3770 y Fn(1)2079 3933 y(A)2158 3766 y(\014)2158 3820 y(\014)2158 3875 y(\014)2158 3930 y(\014)2158 3984 y(\014)2158 4039 y(\014)2214 3952 y FA(6)p FC(=)25 b(0)g FB(;)0 4262 y FC(or)30 b(equiv)-5 b(alen)m(tly)d(,)1480 4341 y FB(\013)1538 4355 y Fy(I)1569 4364 y Fj(1)1628 4341 y FA(\000)20 b FB(\013)1777 4355 y Fy(I)1808 4364 y Fj(2)p 1480 4381 367 4 v 1480 4465 a FB(\013)1538 4479 y Fy(I)1569 4488 y Fj(2)1628 4465 y FA(\000)g FB(\013)1777 4479 y Fy(I)1808 4488 y Fj(3)1881 4402 y FA(6)p FC(=)1987 4341 y FB(\013)2045 4355 y Fy(J)2084 4364 y Fj(1)2143 4341 y FA(\000)g FB(\013)2292 4355 y Fy(J)2331 4364 y Fj(2)p 1987 4381 383 4 v 1987 4465 a FB(\013)2045 4479 y Fy(J)2084 4488 y Fj(2)2143 4465 y FA(\000)g FB(\013)2292 4479 y Fy(J)2331 4488 y Fj(3)2405 4402 y FB(:)0 4627 y FC(Therefore,)30 b(the)h(mapping)e FB(g)34 b FC(is)29 b(one-to-one)j(o)m(v)m(er)g(the)f(domain)2255 4554 y Fn(\000)2297 4587 y Fl(\013)2306 4659 y Fx(3)2351 4554 y Fn(\001)2392 4627 y FC(.)p 3834 4627 4 62 v 3838 4569 55 4 v 3838 4627 V 3892 4627 4 62 v 0 4796 a SDict begin H.S end 0 4796 a 0 4796 a SDict begin 13.6 H.A end 0 4796 a 0 4796 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 0 4796 a 118 x FD(4)135 b(Our)44 b(Deco)t(ding)h(Algorithm)0 5117 y FC(In)35 b([)144 5117 y SDict begin H.S end 144 5117 a 0 TeXcolorgray FC(17)p 0 TeXcolorgray 236 5058 a SDict begin H.R end 236 5058 a 236 5117 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.con2024optimal) cvn H.B /ANN pdfmark end 236 5117 a FC(],)j(the)e(authors)g (designed)f(explicit)f([)p FB(n;)15 b FC(2])1712 5131 y Fy(q)1751 5117 y FC(-RS)36 b(Co)s(des,)i(o)m(v)m(er)f(a)g(\014eld)d (size)i(of)h FB(q)h FC(=)c 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