An extension of the Walsh functions onto the Galois fields GF(2(m)) is intr
oduced. Fields with a simple relation between the traditional and extended
Walsh functions are found. For a memoryless channel, an algorithm of the op
timum symbol-by-symbol reception of signals based on linear codes over GF(2
(m)) is synthesized. The algorithm uses a fast spectral transform in the Wa
lsh basis (FWT).