The error-trapping technique, whenever applicable, is easy to implement. Here we investigate the capability of this technique, specially based on the permutation decoding concept. The object is to give exact lower bounds on the code lengthn, for givenk, of the "multiple-error-correcting'' binary(n, k, t)cyclic codes by applying cyclic(T)and squaring(U)(or square rooting) group(T, U)permutations for1)two-step(T, U)permutation decodable codes(todd- and even-valued) and2)three-step(T,U)permutation decodable codes(todd-valued andt = 2). Finally, some general results are presented for the codes that are not permutation decodable for the specific(T, U)group permutations. The derivation of the results involves only the symbol positions of the errors, and consequently, the results are directly applicable to cyclic codes over GF(2^{m}).

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd \geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = \{0, 1, \alpha , \alpha^{2} , \cdots , \alpha^{n(q-1)-1} \}, the generator polynomialg(X), ofV, has\alpha^{\nu}as a root if and only if\nuhas the form\nu = i(q - 1)and\max_{0 \leq l < s} W_{q}(2^{l} \nu) \leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbers\nu, such that\alpha^{\nu}is a root ofg(X). LetC_{1}, C_{2}, \cdots, C_{\nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.

Adaptive quantizers for PCM and DPCM coding of speech have been of great interest, and several excellent quantization strategies have evolved [1], [2]. In this paper we propose another algorithm for the step-size change in a quantizer. In the present strategy the quantizer is always a linear one except for the fact that the step size can be changed by a fixed amount at each sampling instant. This fixed value is also the minimum allowed step size. Hence, we have the name incremental adaptive quantizer. The quantizer has been simulated on a computer and its performance has been compared with that of Jayant's adaptive quantizer (JAQ), both in PCM and DPCM coders. The figure of merit used in the comparison is the signal-to-noise ratio (SNR) with a correlated sequence as the input. The preliminary results indicate that the new design can perform very well. However, it has the drawback of not being able to respond to very fast changes in the input. It can be easily realized in an integrated form using ROM's.

Certain useful properties of the cyclic codeVare discussed with wordsV(x)=V_{1}(x)(l+x^{n})/(l+x^{n_{1}})+V_{2}(x)(l+x^{n})/(l+x^{n_{2}}), where fori=1,2,V_{i}(x)belongs to a binary codeV_{i}of lengthn_{i}.

This correspondence presents expressions for the Mean Error-Free Interval in a packet-oriented data transmission system. In particular, the case of transmission through a random-error channel is studied, with some results also given for more general noisy channels.

LetV\primebe a binary(n,k)majority-logic decodable code withg\prime (X)as its generator polynomial and odd minimum distanced. LetVbe the(n, k - 1)subset code generated byg\prime (X)(1 + X). This correspondence shows thatVis majority-logic deeodable withd + 1orthogonal estimates. This fact is useful in the simultaneous correction of random errors and erasures.

Some optimal rate-\frac{1}{2}quasi-cyclic codes found by Chert et aL [3] are grouped into a small set of equivalence classes with the assistance of a computer. The weight distribution of a selection of these rate-\frac{1}{2}codes is tabulated. In addition, a list of new optimal rate-\frac{2}{3}quasi-cyclic codes of lengths up to 54 is presented.

In a recent note [1] an analysis of certain binary double-error-correcting cyclic codes was made from the point of view of permutation decodability. Specifically, upper bounds on the rates of these codes were established so that the codes would be permutation decodable. The object of the present correspondence is to give corresponding results for binary triple-error-correcting cyclic codes of certain lengths.

This paper considers the problem of designing an n-input, n-output, asynchronous unit delay (AUD) [2], [7], [13]. In the general case, all input changes are allowed in an AUD. However, the restricted case where only single input changes are allowed has been investigated in detail. Starting with linear single error-correcting codes, an unusual method of obtaining a uniquely reduced flow table of a restricted AUD is developed. Such a reduced table has 2k substitution property (SP) partitions [15] on the set of internal states whose product is the zero partition, where k is the smallest integer equal to or greater than log2 n. Hence, the state behavior of an n-input n-output restricted AUD is realizable by 2k two-state sequential circuits connected in parallel [15]. A direct algorithm for obtaining this realization is also given.

Stone^1found that multiple-error-correcting codes inherently have the ability to correct multiple bursts. Using his methods, somewhat stronger theorems are derived here, and a decoding procedure is given.