CSS-Like Constructions of Asymmetric Quantum Codes

Abstract

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the Fq-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products.After establishing some theoretical foundations, bestperforming CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure q-ary CSS-like codes for q ∈ {2, 3, 4, 5, 7, 8, 9} up to reasonable lengths are found. In many instances, removing the Fq-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.Index Terms-asymmetric quantum codes, best-known linear codes, Delsarte bound, group character codes, cyclic codes, inner products, linear programming bound, quantum Singleton bound, subfield linear codes I. INTRODUCTIONMost of the work to date on quantum error-correcting codes (quantum codes) assumes that the quantum channel is symmetric, i.e., the different types of errors are assumed to occur equiprobably. However, recent papers (see [13] and [20], for instance) argue that in many qubit systems, phase-flips (or Z-errors) occur more frequently than bit-flips (or X-errors). This leads to the idea of adjusting the error-correction to the particular characteristics of the quantum channel and codes M. F. Ezerman was with

Xing–Ling codes, duals of their subcodes, and good asymmetric quantum codes

“…For example, the currently stored [7,2,5] 4 -code in MAGMA does not contain the words of weight 7 but the XL code C 4 (1, 2, 0) listed as Entry 4 in Table 2 contains 1.…”

“…There are also a couple of known instances where the CSS-like construction based on subfield linear codes as discussed in [5] yields AQCs with better parameters than those based on the XL codes. The note at the end of Table 2 highlights two such instances.…”

“…For d x ≥ 4 the derived AQCs have very little, if at all, overlap with previously known ones. The comparison of parameters are done with respect to the list provided in [8] and the more recent results in [5].…”

Vector-Circulant Matrices and Vector-Circulant Based Additive Codes over Finite Fields

“…Since additive codes over finite fields have applications in both classical and quantum communications (see, for example, [1][2][3][4][5]), it is of natural interest to study this family of codes. Constructions of good/optimal additive codes have been widely studied (see [3,[8][9][10], and references therein).…”

“…Coding theory has been introduced to deal with this problem since the 1960s. Additive codes constitute an important class of codes due to their rich algebraic structures and wide applications in both classical and quantum communications (see [1][2][3][4][5], and references therein).…”