Asymptotically dense nonbinary codes correcting a constant number of localized errors [Manuskript]

The binary case was studied in [1], but the method used there doesn’t give the tight answer for nonbinary cases and we presented in [2] another method for the corresponding result. Here we formulate the main theorem and prove the auxiliary statements used in [2]. During the transmission of q–ary words of length n over the channel at most t errors occur, and the encoder knows the set E of t positions, where these errors are possible. The decoder doesn’t know anything about these positions. Let Et = { E | E ⊆ {1, 2, . . . , n}, |E| = t } be the set of all subsets from {1, 2, . . . , n} of size t and let M be a set of messages (|M| = M) . A code word x(m,E) depends not only on the message m ∈ M but also on the configuration of possible errors E . So there exists the natural correspondence between the message m ∈ M and the list of code words ⋃ E∈Et { x(m,E) } , which we use for the transmission of this message. Thus the code X for the set of messages M represents a collection of M lists { ⋃ E∈Et {x(m,E)},m ∈ M } . Sinde we can use the same word for different configurations, the size of a list can be essentially smaller than the size of the set Et ( |Et| = ( n t )) . Let us define the cylinder C(a,A) with the base a = (a, . . . , an) and the support A ( A ⊆ {1, 2, . . . , n} ) as the set of words (y1, . . . , yn) with yi = ai , if i / ∈ A . It is clear that the size of the cylinder C(a,A) is equal to q and the number of different cylinders with the same support A is equal to q . As a result of the transmission of the codeword x(m,E) every word of C ( x(m,E), E ) can appear as output of the channel. The code X corrects t localized errors, if the decoder can correctly recover every message m ∈ M . The following condition is necessary and sufficient for it: C ( x(m,E), E ) ∩ C ( x(m, E), E ) = ∅ for all E,E ∈ Et,m,m ′ ∈ M,m 6= m. (1) Typeset by AMS-TEX 1 The maximal number of messages, which we can transmit by a code correcting t localized errors, is denoted by Lq(n, t) . Proposition 1: Lq(n, t) ≤ q St where St = ∑t i=0(q − 1) C n is the size of a sphere of radius t in the Hamming n–space. A proof of this bound in the q–ary case can be given as for the binary case in [3] or [4]. The key inequality there has the following generalization. Lemma 1. Let C(ai, Ai), . . . , C(aT , AT ) be cylinders with pairwise different supports Ai 6= Aj , i 6= j . Then for the size of the union of the cylinders