Ahlswede R, Bassalygo LA, Pinsker MS. Asymptotically dense nonbinary codes correcting a constant number of localized errors [Manuskript]. Comptes rendus de l' Académie bulgare des Sciences. 1993;46(1):35-37

Ee T

L. A. Bassalygo

M.S. Pinsker

Phi S

R. Ahlswede

English

Ahlswede, Rudolf

Bassalygo, L. A.

Pinsker, M. S.

Publications at Bielefeld University

ASYMPTOTICALLY DENSE NONBINARY CODES CORRECTINGA CONSTANT NUMBER OF LOCALIZED ERRORSR. Ahlswede, L.A. Bassalygo, M.S. PinskerAbstractThe binary case was studied in [1], but the method used there doesn’t give thetight answer for nonbinary cases and we presented in [2] another method for thecorresponding result. Here we formulate the main theorem and prove the auxiliarystatements used in [2].During the transmission of q–ary words of length n over the channel at most terrors occur, and the encoder knows the set E of t positions, where these errorsare 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 sizet and let M be a set of messages (|M| = M) . A code word x(m,E) dependsnot only on the message m ∈ M but also on the configuration of possible errorsE . So there exists the natural correspondence between the message m ∈ M andthe list of code words⋃E∈Et{x(m,E)}, which we use for the transmission of thismessage. Thus the code X for the set of messages M represents a collection ofM lists{⋃E∈Et{x(m,E)},m ∈ M}. Sinde we can use the same word for differentconfigurations, the size of a list can be essentially smaller than the size of the setEt(|Et| =(nt)). Let us define the cylinder C(a,A) with the base a = (a, . . . , an) andthe 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|A| and thenumber of different cylinders with the same support A is equal to qn−|A| .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 thedecoder can correctly recover every message m ∈ M . The following condition isnecessary 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-TEX1The maximal number of messages, which we can transmit by a code correcting tlocalized errors, is denoted by Lq(n, t) .Proposition 1:Lq(n, t) ≤qnStwhere St =∑ti=0(q − 1)iCin is the size of a sphere of radius t in the Hammingn–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 supportsAi 6= Aj , i 6= j . Then for the size of the union of the cylinders∣∣∣∣∣T⋃i=1C(ai, Ai)∣∣∣∣∣ ≥T∑i=1(q − 1)|Ai|.Proof: We proceed by an induction on n . For n = 1 the statement is obvious. LetC(a1, A1), . . . , C(aT , AT ) satisfy the condition of the Lemma. We consider now thenew family C of cylinders:a) if n /∈ Ai , then C(ai, Ai) ∈ Cb) if n ∈ Ai , then C(a(k)i , Ai r n) ∈ C for all k(k = 0, 1, . . . , q − 1) , wherea(k)i = (ai1, . . . , ain−1, k) .We haveC =q−1⋃k=0C(k)where C(k) – all cylinders from C whose last coordinate is equal to k(k = 0, 1, . . . , q−1) . It is easy to show that∣∣∣∣∣⋃C∈CC∣∣∣∣∣ =q−1∑k=o∣∣∣∣∣∣⋃C∈C(k)C∣∣∣∣∣∣ .If follows from the condition of the Lemma that the support Ai r n of q cylindersC(a(k)i , Ai r n), k = 0, 1, . . . , q − 1 differs from the support of other cylinders at leastfor q − 1 subfamilies Ck , k = 0, 1, . . . , q − 1 . Thus one proves the Lemma usingthe induction step to estimate |⋃C∈C(k) C | . It is easy to obtain Proposition 1 fromLemma 1. In fact for every union of cylinders⋃E∈EtC(·, E) there exists some unionof cylinders⋃E∈Sti=0 EiC(·, E) such that2⋃E∈EtC(·, E) =⋃E∈Sti=1 EiC(·, E)and therefore by Lemma 1∣∣∣∣∣⋃E∈EtC(·, E)∣∣∣∣∣ =∣∣∣∣∣∣⋃E∈Sti=1 EiC(·, E)∣∣∣∣∣∣ ≥ St. (2)Now we have from the condition (1) that( ⋃E∈EtC(x(m,E), E))∩( ⋃E′∈EtC(x(m′, E′), E′))= ∅ for m 6= m′.From here and (2) the Proposition follows.The following lower bound can be easily deduced by the standard greedy algorithm(maximal coding).3Proposition 2:Lq(n, t) ≥qnq2t(nt) .Proof: Let X be the code X ={⋃E∈Et{x(m,E)}, m ∈ M}for M messages,correcting t localized errors. As∣∣∣∣∣⋃m∈M⋃E∈EtC(x(m,E), E)∣∣∣∣∣ ≤ M ·(nt)· qtand the number of different cylinders with the same support E′(|E′| = t) is equal toqn−t , for any support E′ ∈ Et there exists a cylinder C(a,E′) withC(a,E′) ∩ C(x(m,E), E)= ∅ for all m ∈M and E ∈ Et,ifM(nt)qt < qn−t. (3)Therefore, if the inequality (3) takes place, it is possible, according to the condition(1), to construct the code for M + 1 messages, correcting t localized errors. HenceProposition 2 follows.Already Proposition 1 and 2 imply the asymptotic equivalence within a constantLq(n, t) ≍qnnt, when t is fixed and n →∞.We draw attention to the fact that this equivalence is known for nonbinary error–correcting codes, except t = 1 , only for t = 2 and q = 3, 4 [5].The following theorem gives the precise constant in the equivalence.Theorem. For every constant tLq(n, t) =qnSt(1 + o(1))=t!(q − 1)t·qnnt(1 + o(1)),where o(1) → o as n →∞ ( o(1) depends certainly on t and q ).When proving this theorem [2] we refered to the following recurrence relation, havingan independent interest.4Lemma 2. If N ≤ (q − 1)T + 1 , thenLq(N + T, 1) ≥ qN−1Lq(T, 1).Proof: On the first N positions we always transmit a parity check ( mod q) codeof the size qN−1 . The last T positions we reserve for the code, which allows us totransmit Lq(T, 1) messages and to correct a single localized error. The method oftransmission depends on the position of the localized error in the following way:a) The error is on the last T positions. We use our code as a code, correcting asingle localized error.b) The error is in the first N positions. We use our code for the transmission ofboth, the message and the number of the position, where the error can occur.According to Proposition 1 and Lemma 2 at least (q − 1)T + 1 different wordsin the output of the channel of length T correspond to everyone of Lq(T, 1)messages and therefore we can make the successful transmission, ifN ≤ (q − 1)T + 1.The proof is complete.References1. L.A. Bassalygo, S.I. Gelfand, M.S. Pinsker, Simple methods of deduction of lowerbounds in coding theory, Probl. Peredaci Inform., vol. 27, 1991, no. 4, 3–8.2. R. Ahlswede, L.A. Bassalygo, M.S. Pinsker, Nonbinary codes correcting localizederrors, IEEE Inf. Theory, in print.3. L.A. Bassalygo, S.I. Gelfand, M.S. Pinsker, Coding for channels with localizederrors, Proc. 4–th Soviet–Swedish Workshop in Information Theory, Gotland,Sweden, 1989, 95–99.4. L.A. Bassalygo, S.I. Gelfand, M.S. Pinsker, Coding for partially localized errors,IEEE Inf. Theory, vo. 37, no. 3, May 1991, 880–884.5. I.I. Dumer, V.A. Zinoviev, New maximum codes over GF(4), Probl. PeredaciInform., vol. 14, 1978, no. 3, 24–34.5

Coding for channels with localized errors,

Coding for partially localized errors,

New maximum codes over GF(4), Probl.

Nonbinary codes correcting localized errors,

Simple methods of deduction of lower bounds in coding theory,

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

http://pub.uni-bielefeld.de/publication/1780512

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

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