Nonstandard coding method for nonbinary codes correcting localized errors

Abstract

In [l] we claimed that the transmission rate of a nonbinary (q-ary) code of length n which corrects localized errors asymptotically equals the Hamming bound on an interval T E [0, TO], TO 5 1/2, and conjectured that TO = 1 / 2 (transmission rate is zero if the number of errors is greater than or equal to n / 2 ) . Though we have not succeeded in proving our conjecture we decided t o promulgate the derivation of the incomplete result on the Hamming bound, the more so, as the method of the proof itself is of independent interest.The situation with localized errors (when the encoder knows the positions where errors can occur) is characterized by the following fact : to attain the Hamming bound asymptotically, it suffices t o provide the decoder with "small" information. Indeed, let us divide the transmitted segment into a growing number of segments of equal length (the length is also growing) and arrange the segments in an ascending order with respect to the number of possible errors on them. The number of the first segment and the number of a set that covers the positions of possible errors on this segment are precisely the "small" information that should be known to the decoder. Now, let us explain why this information is sufficient to attain the Hamming bound. On the first segment, outside the positions of the covering set, we transmit to the decoder the number of the second segment and the number of a set that covers the error positions on this segment. The remaining positions of the first segment are used for message transmission. On the second segment we do the same procedure, and so on. Surely one can ask why we transmit covering sets but not the actual positions of possible errors, which, in fact, would leave more positions for message transmission. The answer is simple: with the optimal choice of covering sets, the gain due to their rather small number compensates the loss of positions for message transmission. Simple computations show that thus we attain the Hamming bound.Thus, the main problem is to transmit the mentioned "small" information to the decoder. We overcome this problem with the help of special nonstandard encoding and decoding procedures. In encoding, we first construct three auxiliary code words from which the transmitted word is constructed. In decoding, we construct three auxiliary "de-code" words from the word received and then reconstruct the transmitted message using them.Transmission of "small" information is performed with the help of binary constant-weight codes that correct localized errors and defects (see (21). To employ these codes, we proceed as follows : start the above-mentioned encoding procedure not from the first segment but from the ( k + l)st, leaving the first k segments free (the total length of the first k segments is small). Then this encoding procedure generates the first q-ary codeword outside the first k segments and in positions outside the covering sets on the other segments. The unity positions of this q-ary codeword (i.e. the positions with the symbol 1 in the...