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International Journal of Information and Coding Theory

Cyclic double struck capital Fq-linear double struck capital Fqt-codes

W. Cary Huffman

Published Jan 1, 2010• 13 citations
Publisher

Abstract

In Calderbank, self-orthogonal additive codes over double struck capital F4 under the traceinner product were connected to binary quantum codes; a similar connection was given in the non-binary case in Rains. In this paper, we consider a natural generalisation of additive codes to double struck capital Fq-linear double struck capital Fqt-codes. We develop the theory of these codes when they are cyclic and count them. Then, we place two different trace inner products on these codes and decide precisely when the cyclic ones are self-orthogonal under these two inner products. We examine the case t = 2 in detail giving specific bases for the self-orthogonal and self-dual cyclic codes. When t ≥ 2, we present counts for the number of self-orthogonal and self-dual cyclic codes under each of the inner products.

DOI
10.1504/ijicot.2010.032543

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In Calderbank, self-orthogonal additive codes over double struck capital F4 under the traceinner product were connected to binary quantum codes; a similar connection was given in the non-binary case in Rains. In this paper, we consider a natural generalisation of additive codes to double struck capital Fq-linear double struck capital Fqt-codes. We develop the theory of these codes when they are cyclic and count them. Then, we place two different trace inner products on these codes and decide precisely when the cyclic ones are self-orthogonal under these two inner products. We examine the case t = 2 in detail giving specific bases for the self-orthogonal and self-dual cyclic codes. When t ≥ 2, we present counts for the number of self-orthogonal and self-dual cyclic codes under each of the inner products.
On the theory of Fq\mathbb{F}_qFq​-linear Fqt\mathbb{F}_{q^t}Fqt​-codes
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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.
The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.
We give a polynomial representation for additive cyclic codes over $ \mathbb{F}_{p^2} $. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $ \mathbb{F}_p $. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $ \mathbb{F}_{p^2} $. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $ \mathbb{F}_{4} $.
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.