Determination of Optimal H-Matrices for 2-Bit Error Correcting Codes

Abstract

2-bit error correction for random errors in memory cells is of growing importance. Instead of the 1-bit error correcting and 2-bit error detecting Hsiao-code a 2-bit error correcting BCH-code can be used with the disadvantage that for the maximal code length of $2^m - 1$ the number of necessary check bits is $2 \cdot m$. Compared to the needed m check bits for the commonly implemented Hsiao-code the number of check bits is doubled and the necessary overhead for error correction is relatively high. To reduce this overhead it is of interest to determine 2-bit error correcting codes with maximal length for a given number of check bits.

In this paper it is shown for the first time that the code length of a 2-bit error correcting BCH-code cannot be enlarged by adding a further column to its H-matrix. (The proof is based on the fact that a 2-bit error correcting BCH code is quasi-perfect.) A similar result is also true for a 2-bit error correcting and 3-bit error detecting BCH-code with included parity.

For up to 8 check bits H-matrices for codes with maximal code length are determined. For larger numbers of check bits H-matrices with almost optimal code length are determined by a new algorithm of computer search, based on detailed properties of the columns of the corresponding H-matrices in there separated form.