Quantum codes from affine variety codes and their subfield-subcodes

Abstract

Abstract. We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction. With this procedure we get codes with good parameters, some of them exceeding the quantum Gilbert-Varshamov bound given by Feng and Ma.

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

“…Instead of the classical way, our BCH codes are regarded as subfield-subcodes of evaluation codes defined by evaluating univariate polynomials [11]. We consider this construction because it can be extended to evaluation by polynomials in several variables [20,18] which we hope will give better codes in the future.…”

“…For convenience, we will write A = {a 0 = 0 < a 1 < a 2 < · · · } = {a j } z j=0 . We will use the following two results which can be found in [20,18].…”

Locally recoverable J-affine variety codes

“…The defining set of the first one is ∆ = I (0,0) ∪ I (0,1) ∪ I (1,0) ∪ I (1,1) ∪ I (2,0) ∪ I (3,0) . The defining sets of the remaining ones are obtained by successively adding to ∆ the following cyclotomic sets: I (4,0) , I (5,0) , I (6,0) , I (7,0) , I (8,0) , I (9,0) , I (10,0) and I (11,0) . Note that we get codes with better defect D δ−1 than in Table 5.…”

“…Section 1 recalls some basic facts on LRC codes. In Subsection 2.1 we introduce Jaffine variety codes which also gave rise to good quantum error-correcting codes [6,7,4]. Subsections 2.2 and 2.3 show J-affine variety codes for which locality and (r, δ)-locality of their subfield-subcodes can be determined.…”

On the evaluation codes given by simple $$\delta $$ δ -sequences

“…The above procedures do not guarantee decoding up to the actual distance, this can be carried out by using the affine variety code point of view [14,40]. Notice that this point of view is also useful to construct quantum codes [15,16,17].…”