New Binary and Ternary LCD Codes

Abstract

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfieldsubcodes of J-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield-subcodes of J-affine variety codes and provide some new and good LCD codes coming from this construction.

Entanglement-assisted quantum error-correcting codes over arbitrary finite fields

“…These kinds of decomposition arise naturally (i.e., for the usual generators) in some families of evaluation codes as BCH codes, toric codes, J-affine variety codes, negacyclic codes, constacylic codes, etc. and the previous approach has been exploited for constructing stabilizer quantum codes, EAQECCs and LCD codes (see [7,8,9,14,17] for instance).…”

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.…”

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

“…Parameters are showed in Table 8. The defining set of our first code is ∆ = I (0,0) ∪ I (0,1) ∪ I (1,1) ∪ I (2,0) ∪ I (3,0) , and the remaining ones are obtained by successively adding to ∆ the following cyclotomic sets: I (4,0) , I (5,0) , and both I (4,1) and I (6,1) .…”

“…The defining set of our first code is ∆ = I (0,0) ∪ I (0,1) ∪ I (1,0) ∪ I (2,0) . We get the remaining ones by successively adding to ∆ the following cyclotomic sets: I (3,0) , I (4,0) , I (5,0) , both I (1,1) and I (6,0) , and I (7,1) . Example 5.…”