The MacWilliams identity for mm-spotty weight enumerators of linear codes over finite fields

Some of the error control codes are applied to high speed memory systems using RAM chips with either 1-bit I/O data (b=1)(b=1) or 4-bit I/O data (b=4)(b=4). However, modern large-capacity memory systems use RAM chips with 8, 16, or 32 bits of I/O data. A new class of codes called mm-spotty byte error codes provides a good source for correcting/detecting errors in those memory systems that use high-density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). The MacWilliams identity provides the relation of weight distribution of a code and that of its dual code. The main purpose of this paper is to present a version of the MacWilliams identity for mm-spotty weight enumerators of linear codes over arbitrary finite fields.