The square @math of a linear error correcting code @math is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in @math . Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both @math and @math . In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes @math with a relatively large dimension of @math and minimum distance of the square @math are discussed. In some cases, the constructions lead to codes @math such that both @math and @math simultaneously have the largest possible minimum distances for their length and dimensions.
We characterize product-maximum distance separable (PMDS) pairs of linear codes, i.e., pairs of codes @math and @math whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions @math and @math . We prove in particular, for @math , that if the square of the code @math has minimum distance at least 2, and @math is a PMDS pair, then either @math is a generalized Reed–Solomon code, or @math is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics. We give an upper bound that relates the dimensions of some given number of linear codes, with the minimum distance of their componentwise product. A typical result is as follows: given t linear codes Ci of parameters [n,ki]q with full support, one can find codewords ci ∈ Ci such that 1 ≤ w(c1*⋯*ct) ≤ max(t-1, n+t-(k1+⋯+kt)).