Some codes related to BCH-codes of low dimension

We construct a large number of record-breaking binary, ternary and quaternary codes. Our methods involve the study of BCH-codes over larger fields, concatenation, construction X and variants of the Griesmer construction (residual codes). 1 Review of the theory Let IFq be the ground field, F = IF q 2. Denote the interval {i, i + 1,..., j} ⊂ ZZ/(q 2 −1)ZZ by [i, j]. Let A = [i, j] ⊂ ZZ/(q 2 −1)ZZ. Interval A determines a (primitive) BCH-code C(A) of length q 2 −1 and minimum distance ≥ | A | +1. The dimension of C(A) is determined with the help of cyclotomic cosets. The cyclotomic coset containing i ∈ ZZ/(q 2 − 1)ZZ is Z(i) = {i, qi} ⊂ ZZ/(q 2 − 1)ZZ. We have | Z(i) | = 1 if and only if (q + 1) | i, | Z(i) | = 2 otherwise. ZZ/(q 2 − 1)ZZ is the disjoint union of ¿the different cyclotomic cosets. The 1 dimension of C(A) is the sum of the cardinalities of the cyclotomic cosets, which intersect A trivially.