Further results on the semilinear equivalence of linear codes
Introduction
The semilinear equivalence of codes is a generalization of a basic concept, code equivalence, in coding theory. The well-known MacWilliams theorem [12], [13] established an equivalent condition for code equivalence, which shows that any isomorphism between two linear codes and preserving the Hamming weight induces an equivalent relation of and , where “isomorphism” is a one-to-one mapping preserving linearity between two vector spaces (here linear codes are viewed as vector spaces). Although the computational complexity of the code equivalence problem is at least as hard as graph isomorphism [6], [15], [14], code equivalence and the MacWilliams theorem have yet been extensively studied because of the justification in theoretical meaning (see [1], [19], [21], [22]). Making use of the generalized Hamming weights (GHWs) [20], Fan et al. [4] established a different code equivalence, which generalized the MacWilliams theorem.
Recently, some different sufficient conditions of equivalence and semilinear equivalence of codes were introduced in [8], [9], [10] through the application of the relative generalized Hamming weights (RGHWs) [11]. In this paper, we will proceed further in this direction and address a more concise equivalent condition for semilinear equivalence of codes. The main result of the present paper substantially improves what was derived in [10] and includes the result in [8] as a special case. More concretely, the present paper generalizes the computing method used in [9] and gives answers to the common questions remaining unsolved in [8], [10], which will be explained in the later sections in detail.
The remainder of this paper is organized as follows. Section 2 sets up notation and terminology and briefly reviews some basic facts. The main results are stated and proved in Sections 3 Main results, 4 The proofs of the main results, respectively. Section 5 concludes the study.
Section snippets
Preliminaries
Let q be a power of a prime and be a finite field with q elements, and let be the set consisting of all vectors with n coordinates over .
A k-dimensional linear code with length n, usually said to be an code, is a k-dimensional linear subspace of . A subcode of is a subspace of . The support of is defined as the set of positions where not all the codewords of have zero coordinates. In particular, the support of a codeword (or the support of the
Main results
In this section, we present an equivalent condition in Theorem 1 for a one-to-one semilinear mapping to be a semimonomial transformation. The proposed equivalent condition substantially generalizes the ones given in [9], [10]. We also present Theorem 2 as a supplement to Theorem 1. The proofs of Theorem 1, Theorem 2 will be given in Section 4. Theorem 1 Assume that and have the same effective length. Let ϕ be a one-to-one semilinear mapping between and . Then ϕ is a semimonomial transformation if
The proofs of the main results
In this section, we denote a q-ary Gaussian binomial coefficient [17] by
Based on the finite projective geometry method described in Lemma 1, we can finish the proof of Theorem 1. In a completely similar way, Theorem 2 can also be proved by using Proposition 1, and so its proof will be omitted here.
Proof of Theorem 1 To show that ϕ is a semimonomial transformation, it is sufficient to check that for [[10] Lemma
Conclusion
Based on relative generalized Hamming weights, for a linear code and its subcode , we first defined relative subcodes of the pair . We then established a tool to characterize relative subcodes, by using finite projective geometry and relative projective subspaces. Employing this tool, we proved that any one-to-one semilinear mapping between and of the same effective length, which preserves the support weight of each relative subcode, is a semilinear equivalence of
Acknowledgements
The author would like to thank the anonymous referees for valuable remarks and helpful suggestions, and they are indebted to Chunlei Li of the University of Bergen for his help in revising this paper.
References (22)
- et al.
An elementary proof of the MacWilliams theorem on equivalence of codes
Inform. Control
(1978) Classification of Griesmer codes and dual transform
Discr. Math.
(2009)-Extension of linear codes
Discr. Math.
(2009)- et al.
New code equivalence based on relative generalized Hamming weights
Inform. Sci.
(2011) - et al.
Characters and the equivalence of codes
J. Combin. Theory Ser.A
(1996) - et al.
The weight hierarchies of q-ary codes of dimension 4
IEEE Trans. Inform. Theory
(1996) - et al.
Generalized Hamming weights and equivalences of codes
Sci. China (Series A)
(2003) Computing automorphism groups of error-correcting codes
IEEE Trans. Inform. Theory
(1982)- et al.
The relative generalized Hamming weight of linear q-ary codes and their subcodes
Des. Codes Cryptogr.
(2008) - et al.
On the equivalence of linear codes
Appl. Algebra Eng., Commun. Comput.
(2011)
The relative generalized Hamming weight and the semilinear equivalence of codes
Sci. China (Series F)
Cited by (1)
Geometry of semilinear embeddings: Relations to graphs and codes
2015, Geometry of Semilinear Embeddings: Relations to Graphs and Codes
- 1
The work of Z. Liu was supported by the National Natural Science Foundation of China (No. 11171366).
- 2
The work of X. Zeng was supported by the National Natural Science Foundation of China (No. 61170257).