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Advances in Mathematics of Communications

Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order

W. Cary Huffman

Published Jan 1, 2013• 9 citations
Publisher

Abstract

Additive codes over $\mathbb{F}_4$ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic $p$ are connected to codes over $\mathbb{F}_{p^2}$ that are $\mathbb{F}_p$-linear in [30]. These codes that arise as connections with quantum codes are self-orthogonal under a particular inner product. We study a further generalization to codes termed $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are self-dual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case $q=3$ and $t=2$.

DOI
10.3934/amc.2013.7.57

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Self-dual Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes with an automorphism of prime order
Additive codes over F4\mathbb{F}_4F4​ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic ppp are connected to codes over Fp2\mathbb{F}_{p^2}Fp2​ that are Fp\mathbb{F}_pFp​-linear in [30]. These codes that arise as connections with quantum codes are self-orthogonal under a particular inner product. We study a further generalization to codes termed Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are self-dual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case q=3q=3q=3 and t=2t=2t=2.
On the theory of Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes
In [7], self-orthogonal additive codes over F4\mathbb{F}_4F4​ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes. We examine a number of classical results from the theory of Fq\mathbb{F}_qFq​-linear codes, and see how they must be modified to give analogous results for Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.
Self-Orthogonality of qqq-Ary Images of qmq^{m}qm-Ary Codes and Quantum Code Construction
A code over GF can be imaged or expanded into a code over GF using a basis for the extension field over the base field. The properties of such an image depend on the original code and the basis chosen for imaging. Problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory. In this work, a generalized version of the problem of self-orthogonality of the q-ary image of a qm-ary code has been considered. Given an inner product (more generally, a bi-additive form), necessary and sufficient conditions have been derived for a code over a field extension and an expansion basis so that an image of that code is self-orthogonal. The conditions require that the original code be self-orthogonal with respect to several related bi-additive forms whenever certain power sums of the dual basis elements do not vanish. Numerous interesting corollaries have been derived by specializing the general conditions. An interesting result for the canonical or regular inner product in fields of characteristic two is that only self-orthogonal codes result in self-orthogonal images. Another result is that image of a code is self-orthogonal for all bases if and only if trace of the code is self-orthogonal, except for the case of binary images of 4-ary codes. The conditions are particularly simple to state and apply for cyclic codes. To illustrate a possible application, new quantum error-correcting codes have been constructed with larger minimum distance than previously known.
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete classification to codes of length 9 and 10. The classification is obtained by using a new algorithm that combines two graph representations of self-dual additive codes. The search space is first reduced by the fact that every code can be mapped to a weighted graph, and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism. By an extension technique, we are able to classify all optimal codes of length 11 and 12. There are 56,005,876 (11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes with trivial automorphism group.
Additive conjucyclic codes over $ \mathbb{F}_{q^2} $ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $ \mathbb{F}_q $-linear isomorphism between $ \mathbb{F}_{q^2}^n $ and $ \mathbb{F}_q^{2n} $. This correspondence shows that additive conjucyclic codes of length $ n $ correspond bijectively to $ q $-ary linear cyclic codes of length $ 2n $, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $ \mathbb{F}_{q^2}^n $, which is shown to be compatible with the symplectic inner product on $ \mathbb{F}_q^{2n} $ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $ q $-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $ q $-ary QECCs from additive conjucyclic codes, together with an illustrative example.
Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length nnn correspond bijectively to qqq-ary linear cyclic codes of length 2n2n2n, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct qqq-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for qqq-ary QECCs from additive conjucyclic codes, together with an illustrative example.
. In this paper we discuss the status of the classication of the extremal additive codes over GF (4) which are self-dual under the trace inner product. These codes are of interest because of their connection to quantum codes. There are 5 inequivalent codes of length 8, 8 of length 9, at least 56 of length 10, and exactly 1 of length 11. We discuss the methods of classication, the concepts of lengthening and shortening, and connections to binary codes. 1. Introduction An additive code C over GF (4) of length n is an additive subgroup of GF (4) n . As C is a free GF(2)-module, it has size 2 k for some 0 k 2n. We call C an (n; 2 k ) code. It has a basis, as a GF(2)-module, consisting of k basis vectors; a generator matrix of C will be a k n matrix with entries in GF (4) whose rows are a basis of C. Interest in additive codes over GF (4) has arisen because of their correspondence to quantum codes as described in [3]. There is a natural inner product arising from the trace m...