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

Tatsuya Maruta
2003 Discrete Mathematics  
The relation between the 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.  ... 
doi:10.1016/s0012-365x(02)00820-8 fatcat:3mimexgpx5fkjg45hauwkcluwq

Two classes of convolutional codes over GF(q) for q-ary orthogonal signaling

W.E. Ryan, S.G. Wilson
1991 IEEE Transactions on Communications  
Our conclusion is that the most efficient communication system design for M-ary orthogonal channels 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.  ... 
doi:10.1109/26.68274 fatcat:hs7bswhfvbd3dbdfbsy4hprbsm


2018 Journal of Algebra Number Theory Advances and Applications  
The purpose 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  ... 
doi:10.18642/jantaa_7100121864 fatcat:hv7txs4lurahde4ltf33oqvlna

Self-orthogonality of q-ary Images of q^m-ary Codes and Quantum Code Construction [article]

Sundeep B, Andrew Thangaraj
2007 arXiv   pre-print
A 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 ⊥ .  ... 
arXiv:cs/0606106v3 fatcat:tsyllhxrrvdkvfhfrdowbtfgjq

On constant-composition codes over Z/sub q/

Yuan Luo, Fang-Wei Fu, A.J. Han Vinck, Wende Chen
2003 IEEE Transactions on Information Theory  
In this correspondence, we give a lower bound for the maximum size 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  ... 
doi:10.1109/tit.2003.819339 fatcat:5xjuzpmiqfa2lpndqyfwr6xmwm

Sparse Parity-Check Matrices over ${GF(q)}$

2005 Combinatorics, probability & computing  
entries from the finite field 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.  ... 
doi:10.1017/s0963548304006625 fatcat:zf5wi3ozeval7eg4djhir5bd2i

Dimensions of three types of BCH codes over GF(q) [article]

Hao Liu, Cunsheng Ding, Chengju Li
2016 arXiv   pre-print
Furthermore, we settle the 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.  ... 
arXiv:1608.03027v1 fatcat:sp6wzsd5sbfh5klmqdkmbderk4

Constructions of binary constant-weight cyclic codes and cyclically permutable codes

Nguyen Q. A, L. Gyorfi, J.L. Massey
1992 IEEE Transactions on Information Theory  
ACKNOWLEDGMENT The authors are grateful to C. van Pul 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.  ... 
doi:10.1109/18.135636 fatcat:362hqdixr5fqxmln2hxme4uv7a

An infinite family of Steiner systems S(2, 4, 2^m) from cyclic codes [article]

Cunsheng Ding
2017 arXiv   pre-print
The objective 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 q1 < d.  ... 
arXiv:1701.05965v1 fatcat:efjagzhy5bgmpdp23iqsdhehcm

Another Generalization of the Reed-Muller Codes [article]

Cunsheng Ding, Chunlei Li, Yongbo Xia
2017 arXiv   pre-print
The punctured binary Reed-Muller 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 (q1) i . The minimum distance dof ✵(q, m, h) ⊥ is lower bounded by d ⊥ ≥ q m−h + q − 2.  ... 
arXiv:1605.03796v2 fatcat:vdtofrnmdvd3bcaiqfu5znwibq

Optimal Binary Linear Codes from Maximal Arcs [article]

Ziling Heng, Cunsheng Ding, Weiqiong Wang
2020 arXiv   pre-print
The binary Hamming 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.  ... 
arXiv:2001.01049v1 fatcat:ufvpjdbhd5d2lnskg4sodkj3kq

Maximal arcs and extended cyclic codes [article]

Stefaan De Winter, Cunsheng Ding, Vladimir D. Tonchev
2017 arXiv   pre-print
It is proved that for every 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 + 1d] and nonzero weights n + 1d and n + 1.  ... 
arXiv:1712.00137v1 fatcat:34sdt6hmcndkfknoecfidmlgrq

Extendability of 3-weight (mod q) linear codes over Fq

Tatsuya Maruta, Kei Okamoto
2009 Finite Fields and Their Applications  
We consider 3-weight (mod 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.  ... 
doi:10.1016/j.ffa.2008.09.003 fatcat:2wzgivaxxvhezdm6ib3dnnuttm

Notes on Reed-Muller Codes [article]

Yanling Chen, Han Vinck
2009 arXiv   pre-print
For the first order RM 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  ... 
arXiv:0901.2062v2 fatcat:4zh6i47srbchdoi4b4m4eizavy

Hadamard full propelinear codes of type Q; rank and kernel

J. Rifà, Emilio Suárez Canedo
2017 Designs, Codes and Cryptography  
Acknowledgements Acknowledgements The authors are grateful to the anonymous referees for their helpful comments, which have improved the presentation 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.  ... 
doi:10.1007/s10623-017-0429-2 fatcat:kj72v6m5pjhwbb7j62gcrwlh5m
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