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(c
₁
*⋯*c
_t
) ≤ max(t-1, n+t-(k
₁
+⋯+k
_t
)).