An Upper Bound of Singleton Type for Componentwise Products of Linear Codes

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 C_i of parameters [n,k_i]_q with full support, one can find codewords c_i ∈ C_i such that 1 ≤ w(c₁*⋯*c_t) ≤ max(t-1, n+t-(k₁+⋯+k_t)).

I. Introduction

Let be a prime power, and the field with elements. For any integer , let ∗ denote componentwise multiplication in the vector space , so

$$(x_{1},\ldots,x_{n})\ast (y_{1},\ldots,y_{n})=(x_{1}y_{1},\ldots,x_{n}y_{n}).$$ If are linear codes of the same length , let $$C_{1}\ast\cdots\ast C_{t}=\sum_{c_{i}\in C_{i}}{\BBF_{q}}\cdot c_{1}\ast\cdots\ast c_{t}\;\subseteq\,({\BBF_{q}})^{n}$$ be the linear code spanned by the componentwise products of their codewords. (In [8], this was denoted with brackets meant to emphasize that the linear span is taken. Here we will keep notation lighter. All codes in this text will be linear.)