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Extendability of linear codes over GF(q) with minimum distance d, gcd(d,q)=1

2003
*
Discrete Mathematics
*

The relation between the

doi:10.1016/s0012-365x(02)00820-8
fatcat:3mimexgpx5fkjg45hauwkcluwq
*extendability**of**linear**codes**over**GF*(*q*) having the*minimum**distance**d**with**gcd*(*d*;*q*) =*1*and blocking sets*with*respect to lines in the projective space is given. ... From this geometrical point*of*view, some new conditions for which such*codes*are*extendable*are given. ... Acknowledgements This work was completed while the author was visiting the The School*of*Sciences at the University*of*Salford. ...##
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Two classes of convolutional codes over GF(q) for q-ary orthogonal signaling

1991
*
IEEE Transactions on Communications
*

Our conclusion is that the most efficient communication system design for M-ary orthogonal channels

doi:10.1109/26.68274
fatcat:hs7bswhfvbd3dbdfbsy4hprbsm
*with*noncoherent reception would employ low-rate*codes**over**GF*(*q*)*with*M =*q*and*q*> 2. ... standard free Hamming*distance**of*the*code*, N,( w ) is the total number*of*nonzero information symbols attached to remergent trellis paths*with*Hamming codeword weight w.l A listing*of*N,( w ) provides ... Lemma 2: Let g ( x ) generate a cyclic*code**over*F,*with**minimum**distance**d*and length N. ...##
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NEW SELF-DUAL CODES OVER GF (q)

English

2018
*
Journal of Algebra Number Theory Advances and Applications
*

English

The purpose

doi:10.18642/jantaa_7100121864
fatcat:hv7txs4lurahde4ltf33oqvlna
*of*this paper is to studies*codes*C*over*( ) we also generalize some*of*the results obtained by Mac Williams, Odlyzko, Sloane and Ward. ... Wolfmann and all the members*of*G.R.I.M.*of*Toulon for their constructive remarks. Further, I would also like to thank the anonymous referee for valuable comments on the submitted manuscript. ... A*linear**code*C*over*F*of*length n and dimension k consists*of*k 2*q*vectors ( , 0 u u = ) . , ,*1*F u u i n ∈ −*q*3 The weight*of*, u denoted by ( ) u wt is the number*of*nonzero . i u The*minimum**distance*...##
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Self-orthogonality of q-ary Images of q^m-ary Codes and Quantum Code Construction
[article]

2007
*
arXiv
*
pre-print

A

arXiv:cs/0606106v3
fatcat:tsyllhxrrvdkvfhfrdowbtfgjq
*code**over**GF*(*q*^m) can be imaged or expanded into a*code**over**GF*(*q*) using a basis for the extension field*over*the base field. ... To illustrate a possible application, new quantum error-correcting*codes*have been constructed*with*larger*minimum**distance*than previously known. ... We state the theorem found in [*1*] for completeness: Theorem*1*(Calderbank et al [*1*] ): Suppose C is a (n, k)*linear**code**over**GF*(4) where*d*⊥ is the*minimum**distance**of*C ⊥ . ...##
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On constant-composition codes over Z/sub q/

2003
*
IEEE Transactions on Information Theory
*

In this correspondence, we give a lower bound for the maximum size

doi:10.1109/tit.2003.819339
fatcat:5xjuzpmiqfa2lpndqyfwr6xmwm
*of*the -ary constant-composition*codes**with**minimum**distance*at least 3. ... In addition, three construction methods*of*constant-composition*codes*are presented, and a number*of*optimum constant-composition*codes*are obtained by using these constructions. ... By using Theorem 3, we obtain an optimum constant-composition*code**over**GF*(*q*)*with*length*q*k 0*1*, size*q*k 0*1*,*minimum**distance**q*k01 (*q*0*1*), and constant composition [*q*k01 0*1*;*q*k01 ; . . . ;*q*...##
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Sparse Parity-Check Matrices over ${GF(q)}$

2005
*
Combinatorics, probability & computing
*

entries from the finite field

doi:10.1017/s0963548304006625
fatcat:zf5wi3ozeval7eg4djhir5bd2i
*GF*(*q*) and each k columns are linearly independent*over**GF*(*q*). ... For k = 2 i we show that N*q*(m, k, r) = Θ(m kr/(2(k−*1*)) ) if*gcd*(k −*1*, r) = k −*1*, while for arbitrary even k ≥ 4*with**gcd*(k −*1*, r) =*1*we have N*q*(m, k, r) = Ω(m kr/(2(k−*1*)) · (log m)*1*/(k−*1*) ). ... This observation can be used to*extend*the length*of*a*linear**code*, but at the same time we reduce its*minimum**distance*. ...##
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Dimensions of three types of BCH codes over GF(q)
[article]

2016
*
arXiv
*
pre-print

Furthermore, we settle the

arXiv:1608.03027v1
fatcat:sp6wzsd5sbfh5klmqdkmbderk4
*minimum**distances**of*some primitive BCH*codes*. We also explore the dimensions*of*the BCH*codes**of*lengths n=(*q*^m-*1*)/(*q*-*1*) and n=*q*^m+*1**over*finite fields. ... In this paper, we study the dimensions*of*BCH*codes**over*finite fields*with*three types*of*lengths n, namely n=*q*^m-*1*, n=(*q*^m-*1*)/(*q*-*1*) and n=*q*^m+*1*. ... An [n, k,*d*]*linear**code*C*over**GF*(*q*) is a*linear*subspace*of**GF*(*q*) n*with*dimension k and*minimum*(Hamming)*distance**d*. ...##
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Constructions of binary constant-weight cyclic codes and cyclically permutable codes

1992
*
IEEE Transactions on Information Theory
*

ACKNOWLEDGMENT The authors are grateful to C. van Pul

doi:10.1109/18.135636
fatcat:362hqdixr5fqxmln2hxme4uv7a
*of*Philips Crypt0 in Eindhoven, The Netherlands, both for communicating to them the improved lower bound on A( n,*d*, w) given in Section III-C and ... for suggesting the use*of*the Legendre sequences in Constructions III and IV. ...*minimum**distance**d*= n -k +*1*, i.e., an MDS*code*. ...##
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An infinite family of Steiner systems S(2, 4, 2^m) from cyclic codes
[article]

2017
*
arXiv
*
pre-print

The objective

arXiv:1701.05965v1
fatcat:efjagzhy5bgmpdp23iqsdhehcm
*of*this paper is to present an infinite family*of*Steiner systems S(2, 4, 2^m) for all m ≡ 2 4≥ 6 from cyclic*codes*. ... This may be the first*coding*-theoretic construction*of*an infinite family*of*Steiner systems S(2, 4, v). As a by-product, many infinite families*of*2-designs are also reported in this paper. ... Theorem*1*(Assmus-Mattson Theorem). Let C be a [v, k,*d*]*code**over**GF*(*q*). Let*d*⊥ denote the*minimum**distance**of*C ⊥ . Let w be the largest integer satisfying w ≤ v and w − w +*q*− 2*q*−*1*<*d*. ...##
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Another Generalization of the Reed-Muller Codes
[article]

2017
*
arXiv
*
pre-print

The punctured binary Reed-Muller

arXiv:1605.03796v2
fatcat:vdtofrnmdvd3bcaiqfu5znwibq
*code*is cyclic and was generalized into the punctured generalized Reed-Muller*code**over*(*q*) in the literature. ... The major objective*of*this paper is to present another generalization*of*the punctured binary Reed-Muller*code*. ... The dual*code*✵(*q*, m, h) ⊥ has parameters [*q*m −*1*, k ⊥ ,*d*⊥ ], where k ⊥ = h ∑ i=*1*m i (*q*−*1*) i . The*minimum**distance**d*⊥*of*✵(*q*, m, h) ⊥ is lower bounded by*d*⊥ ≥*q*m−h +*q*− 2. ...##
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Optimal Binary Linear Codes from Maximal Arcs
[article]

2020
*
arXiv
*
pre-print

The binary Hamming

arXiv:2001.01049v1
fatcat:ufvpjdbhd5d2lnskg4sodkj3kq
*codes**with*parameters [2^m-*1*, 2^m-*1*-m, 3] are perfect. Their*extended**codes*have parameters [2^m, 2^m-*1*-m, 4] and are*distance*-optimal. ... The second objective is to construct a class*of**distance*-optimal binary*codes**with*parameters [2^m+2, 2^m-2m, 6]. Both classes*of*binary*linear**codes*have new parameters. ... An [n, k,*d*]*code*C*over**GF*(*q*) is a k-dimensional subspace*of**GF*(*q*) n*with**minimum*(Hamming)*distance**d*. The information rate*of*C is defined as k/n. ...##
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Maximal arcs and extended cyclic codes
[article]

2017
*
arXiv
*
pre-print

It is proved that for every

arXiv:1712.00137v1
fatcat:34sdt6hmcndkfknoecfidmlgrq
*d*> 2 such that*d*-*1*divides*q*-*1*, where*q*is a power*of*2, there exists a Denniston maximal arc A*of*degree*d*in (2,*q*), being invariant under a cyclic*linear*group that fixes ... arcs and two-weight*codes*. ... Let mk ≥*1*, and let C (*q*,2,n) be a*linear**code**over**GF*(*q*)*with*parameters [n +*1*, 3, n +*1*−*d*] and nonzero weights n +*1*−*d*and n +*1*. ...##
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Extendability of 3-weight (mod q) linear codes over Fq

2009
*
Finite Fields and Their Applications
*

We consider 3-weight (mod

doi:10.1016/j.ffa.2008.09.003
fatcat:2wzgivaxxvhezdm6ib3dnnuttm
*q*) [n, k,*d*]*q**codes**with**d*≡ −*1*(mod*q*) whose weights are congruent to 0 or ±*1*(mod*q*). ... The latter is a generalization*of*the result on the*extendability**of*ternary*linear**codes*[T. Maruta,*Extendability**of*ternary*linear**codes*, Des. ... Acknowledgments The authors thank the anonymous referees for their careful reading and valuable suggestions, which led to a considerable improvement*of*the original text. ...##
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Notes on Reed-Muller Codes
[article]

2009
*
arXiv
*
pre-print

For the first order RM

arXiv:0901.2062v2
fatcat:4zh6i47srbchdoi4b4m4eizavy
*code*, we prove that it is unique in the sense that any*linear**code**with*the same length, dimension and*minimum**distance*must be the first order RM*code*; For the second order RM*code*... Furthermore, we show that the specified sub-*codes**of*length <= 256 have*minimum**distance*equal to the upper bound or the best known lower bound for all*linear**codes**of*the same length and dimension. ... Note that*d*+ is the upper bound*of*the*minimum**distance*for all the*linear**codes**of*the same length and dimension;*d*− is the largest*minimum**distance*,*of*which a*linear**code**with*the same length and ...##
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Hadamard full propelinear codes of type Q; rank and kernel

2017
*
Designs, Codes and Cryptography
*

Acknowledgements Acknowledgements The authors are grateful to the anonymous referees for their helpful comments, which have improved the presentation

doi:10.1007/s10623-017-0429-2
fatcat:kj72v6m5pjhwbb7j62gcrwlh5m
*of*the results*of*this paper. ... Let C be a binary*linear**code**of*length n and*minimum**distance**d*. The*extended**code*C, is dened as C = {(x*1*, . . . , x n+*1*) ∈ Z n+*1*2 : (x 2 , . . . , x n+*1*) ∈ C*with*n+*1*i=*1*x i = 0}. ... When C is a*linear**code*, then it is known that*d*(C) = wt(C). A binary (n, M,*d*)-*code*is a*code**with*length n, M codewords and*minimum**distance**d*. ...
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