A memory system according to the embodiment comprises a p-adic number converter unit operative to convert δ-digit, h-bit symbols to a k-digit, p-adic data word (p is a prime of 3 or more); an encoder unit operative to generate, from the p-adic data word, a code C composed of a residual field Zp of the prime p; a memory unit operative to store the code C as write data; an error correcting unit operative to apply an operation using a syndrome S generated from read data Y for error correcting the read data Y to regenerate the code C; a decoder unit operative to reverse-convert the code C to regenerate the p-adic data word; and a binary converter unit operative to convert the data word to a binary number to regenerate the binary data D.