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arXiv:1710.02160 · IEEE Transactions on Information Theory

Classical and Quantum Evaluation Codes at the Trace Roots

Carlos Galindo, Fernando Hernando, Diego Ruano

Published Sep 3, 2018• 12 citations
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Abstract

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F4which are records at http://codetables.de/.

DOI
10.1109/tit.2018.2868442

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We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F4which are records at http://codetables.de/.
Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature.
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,the necessary and sufficient condition is proved under which classical convolutional codes over GF(4) are self-orthogonal with respect to the trace inner product,and then corresponding codes satisfying the condition are found.Tail-biting codes with a simple block structure are obtained by tail-biting transformation,which are self-orthogonal with respect to the trace inner product.Finally,relative quantum stabilizer codes are constructed.The code construction is simple,and the minimum distance of the code approaches the upper bound.
Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2q^2q2 q 2 -ary linear codes. This result can be generalized to q2mq^{2 m}q2m q 2 m -ary linear codes, m>1m > 1m>1 m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with q≠2q \ne 2q=2 q ≠ 2 , improving others in the literature.
In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes.
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.
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,one method for constructing quantum constacyclic codes is presented by finding self-orthogonal classical constacyclic codes over the field GF(4) under a trace inner product.Finally,some examples are given.As an application of this method,quantum Hamming codes are constructed.This class of quantum codes includes well-known quantum cyclic codes.