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Extendability of linear codes over GF(q) with minimum distance d, gcd(d,q)=1
2003
Discrete Mathematics
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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
1991
IEEE Transactions on Communications
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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
NEW SELF-DUAL CODES OVER GF (q)
English
2018
Journal of Algebra Number Theory Advances and Applications
English
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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]
2007
arXiv
pre-print
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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/
2003
IEEE Transactions on Information Theory
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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
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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]
2016
arXiv
pre-print
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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
1992
IEEE Transactions on Information Theory
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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]
2017
arXiv
pre-print
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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 q − 1 < d. ...
arXiv:1701.05965v1
fatcat:efjagzhy5bgmpdp23iqsdhehcm
Another Generalization of the Reed-Muller Codes
[article]
2017
arXiv
pre-print
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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 (q − 1) i . The minimum distance d ⊥ of ✵(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]
2020
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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 + 1 − d] and nonzero weights n + 1 − d and n + 1. ...
arXiv:1712.00137v1
fatcat:34sdt6hmcndkfknoecfidmlgrq
Extendability of 3-weight (mod q) linear codes over Fq
2009
Finite Fields and Their Applications
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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]
2009
arXiv
pre-print
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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
2017
Designs, Codes and Cryptography
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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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