Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Advances in Mathematics of Communications

On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

W. Cary Huffman

Published Jan 1, 2013• 18 citations
Publisher

Abstract

In [7], self-orthogonal additive codes over $\mathbb{F}_4$ 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 $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. We examine a number of classical results from the theory of $\mathbb{F}_q$-linear codes, and see how they must be modified to give analogous results for $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-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.

DOI
10.3934/amc.2013.7.349

Related papers

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

W. Cary Huffman

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 $t=2$.

Advances in Mathematics of Communications•Jan 1, 2013•9 citations

On Additive GF(4) Codes

Philippe Gaborit, W. Cary Huffman, Jon-Lark Kim +1 more

. 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...

•Jan 1, 2001•29 citations

Quantum Error Correction via Codes over GF(4)

A.R. Calderbank, Eric M. Rains, Peter W. Shor +1 more

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

arXiv:quant-ph/9608006•Aug 6, 1996•6 citations

Quantum error correction via codes over GF(4)

A.R. Calderbank, Eric M. Rains, Peter W. Shor +1 more

The problem of finding quantum error correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

IEEE Transactions on Information Theory•Jul 1, 1998•1,748 citations

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

W. Cary Huffman

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.

International Journal of Information and Coding Theory•Jan 1, 2010•13 citations

Self-Orthogonality of $q$-Ary Images of $q^{m}$-Ary Codes and Quantum Code Construction

B Sundeep, Andrew Thangaraj

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.

arXiv:cs/0606106•IEEE Transactions on Information Theory•Jun 28, 2007•10 citations
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-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.
. 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...
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.
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.