Projections of Binary Linear Codes onto Larger Fields (original) (raw)
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Optimal linear codes of dimension 4 over GF (5)
Applied Algebra, Algebraic Algorithms and Error- …, 1997
Let nq(k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over the Galois field GF (q). An [n, k, d]-code whose length is equal to nq(k, d) is called optimal. In this paper we present some matrix generators for the family of optimal [n, 3, d] codes over GF (7) and GF (11). Most of our given codes in GF (7) are nonisomorphic with the codes presented before. Our given codes in GF (11) are all new.
Optimal binary codes derived from \mathbb {F}_{2} \mathbb {F}_4$$-additive cyclic codes
Journal of Applied Mathematics and Computing, 2020
In this paper, we study the algebraic structure of additive cyclic codes over the alphabet F r 2 × F s 4 = F r 2 F s 4 , where r and s are non-negative integers, F 2 = GF(2) and F 4 = GF(4) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for F 2 F 4-additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over F 2. Further, we investigate the duals of F 2 F 4-additive cyclic codes. We show that the dual of any F 2 F 4-additive cyclic code is another F 2 F 4-additive cyclic code. Using the mapping W , we provide examples of F 2 F 4-additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over F 4 and give some examples of optimal parameter quantum codes over F 4. Keywords F 2 F 4-additive cyclic codes • Duality • Quantum codes • Optimal codes Mathematics Subject Classification 94B05 • 94B60
New Linear Codes over GF(3)GF(3)GF(3), GF(11)GF(11)GF(11), and GF(13)GF(13)GF(13)
Journal of Algebra Combinatorics Discrete Structures and Applications
Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets F3, F11, and F13 with better parameters. A total of 38 new linear codes over the three alphabets are presented.
2021
In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector a in Fn 2 . We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if C is a right polycyclic code induced by a vector a ∈ Fn 2 , then the Hermitian dual of C is a sequential code induced by a. As an application of these codes, we present examples of additive right polycyclic codes over F4 with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over F4. 1. Preliminaries A linear code of length n over a finite field F is a subspace of F. An Additive code of length n over a finite field F is a subgroup of F. Additive codes over the finite field F4 = { 0, 1, α, α } where α + α + 1 = 0 were introduced in [3] because of their applications in quantum computing. Define t...
Additive Polycyclic Codes over F4 Induced by Binary Vectors and Some Optimal Codes
2021
In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector a in Fn 2 . We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if C is a right polycyclic code induced by a vector a ∈ Fn 2 , then the Hermitian dual of C is a sequential code induced by a. As an application of these codes, we present examples of additive right polycyclic codes over F4 with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over F4. 1. Preliminaries A linear code of length n over a finite field F is a subspace of F. An Additive code of length n over a finite field F is a subgroup of F. Additive codes over the finite field F4 = { 0, 1, α, α } where α + α + 1 = 0 were introduced in [3] because of their applications in quantum computing. Define t...
On codes over 𝓡k, m and constructions for new binary self-dual codes
Mathematica Slovaca, 2016
In this work, we study codes over the ring R k,m = F 2 [u, v]/ u k , v m , uv − vu , which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from R k,m to F km 2 together with a Lee weight. After proving the MacWilliams identities for codes over R k,m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over R k,m. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72,36,12] and 105 new Type II binary self-dual codes of parameter [72,36,12].
Optimal linear codes of dimension 4 over F
be the smallest integer n for which there exists a linear code of length n, dimension IC, and minimum distance d, over a field of q elements. In this correspondence we determine n5 (4, d ) for all but 22 values of d. Index Terms-Optimal q-ary linear codes, minimum-length bounds. Publisher Item Identifier S 0018-9448(97)00108-9.
Quasi-cyclic codes over Z/sub 4/ and some new binary codes
IEEE Transactions on Information Theory, 2002
Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.