If R = F/sub q/[x/spl rceil/]/(x/sup m/ - 1), S = F/sub qn/[x]/(x/sup m/ - 1), we define the mapping a_(x) /spl rarr/ A(x) =/spl sigma//sub 0//sup n-1/a/sub i/(x)/spl alpha//sub i/ from R/sup n/ onto S, where (/spl alpha//sub 0/, /spl alpha//sub i/,..., /spl alpha//sub n-1/) is a basis for F/sub qn/ over F/sub q/. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F/sub q/ of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x/sub m/ 1 over F/sub q/ and F/sub qn/ are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F/sub qn/ over F/sub q/, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.

In this correspondence we apply a recent inequality by de Caen (1997) to derive a lower bound on the probability of error for M-ary signals derived from a binary linear code and used on the additive white Gaussian noise channel with a maximum-likelihood decoder. This bound depends only on the weight enumerator of the code and the signal-to-noise ratio E/sub b//N/sub 0/. We show that this bound converges to the union upper bound as E/sub b//N/sub 0/ goes to infinity. Finally, by means of examples, we compare our lower bound with those of Shannon and Swaszek and with Poltyrev's upper bound.

For (n, q)=1 V a q/sup m/-ary cyclic code of length n and with generator polynomial g(x), we show that there exists a basis for F(q/sup m/) over F/sub q/ with respect to which the q-ary image of V is cyclic, if and only if: (i) g(x) is over F/sub q/; or (ii) g(x)=g/sub 0/(x)(x-/spl gammasup -q(/spl mu/)/), g/sub 0/(x) is over F/sub q/, F/sub qspl ne/F(q/sup k/)=F/sub q/(/spl gamma/)/spl sub/F(q/sup m/), /spl mu/ an integer modulo k, and w/sup m/-/spl gamma/ has a divisor over F(q/sup k/) of degree e=m/k; or (iii) g(x)=g/sub 0/(x) /spl Pisub /spl muspl epsiv/s/(x-/spl gamma/(-q/sup /spl mu)), g/sub 0/(x) is over F/sub q/, F/sub qspl ne/F(q/sup k/)=F/sub q/(/spl gamma/)/spl sub/F(q/sup m/), S a set of integers module k of cardinality k-1 and w/sup m/-/spl mu/ has a divisor over F(q/sup k/) of degree e=m/k. In all of the above cases, we determine all of the bases with respect to which the q-ary image of V is cyclic.>

We show that the ensemble average of the block error probability for the ensemble of terminated rate 1/n fixed convolutional codes, used on the binary symmetric channel with a maximum likelihood decoder, is bounded by exp/sub 2/-NE/sub r/(1-K/N), where N=(L+m)n is the block length, L being the message length, K the constraint length, and E/sub r/() is the random coding exponent for block codes. Hence, E/sub r/(1-K/N)>0 for H(p)>

The notion of a q-ary irreducible quasi-cyclic code of block length n and index r is introduced. A trace description of such a code is provided in a fashion similar to the trace description of irreducible cyclic codes. In particular, it is shown that an irreducible quasi-cyclic code of dimension k is completely described by an irreducible cyclic code and r elements from a field of cardinality q/sup k/. Using this fact, a number of binary irreducible quasi-cyclic codes of index 2 are constructed and their weight spectra obtained.>

The development of a 2400-b/s speech digitizer which provides an acceptable level of intelligibility and quality over land mobile satellite channels is described. Performance tests over simulated channels in the UHF band (800 MHz) are presented. The voice digitizer is a linear prediction (LPC) vocoder which uses a channel error correction and concealment procedure tailored to error statistics for a minimum-shift keyed (MSK) downlink to a moving vehicle. The error-handling technique is based on perceptual criteria and utilizes the parametric nature of LPC representation of speech. A single-error-correcting, single-burst-detecting (28, 20) fire code is shown to be the best choice for the application. The intelligibility of the vocoder is measured and compared to the standard LPC-10 algorithm. The major remaining sources of speech quality degradation due to channel errors are determined and ranked.>

We propose a triple error-correcting product code, designed to provide additional error protection for data consisting of 8-bit bytes all having even (or odd) parity (e.g., ASCII characters). A practical decoding algorithm for the code is described.

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.

In this paper we introduce a class of linear codes especially designed to provide additional error protection for data consisting of bytes all having even (or odd) parity (e.g., ASCII characters). The technique consists in adding an overall parity byte computed as a linear function of the information bytes. The linear function is designed such that the resulting codes can correct all single errors and all double errors occurring in distinct information bytes. It is shown that any code which can correct these latter mentioned error patterns has an overall length of at most 37 bytes, and a specific code of length 29 bytes is described. A practical decoding algorithm for the new class of codes is described. Finally, the performance of the codes, when used on the binary symmetric channel, is compared with that of the row-column codes for which the additional parity byte is simply the modulo-2 sum of the information bytes.

For the case whenkdividesn, we introduce a special class of(n,k)F-ary convolutional codes,F=GF(q)a finite field, by considering the input to an(n,k)encoder as a sequence over GF(q^{k}), the output as a sequence over GF(q^{n})(an idea first used by Dym [10]), and then considering encoders which correspond to convolving the input with a fixed sequence\Gamma_{0}, \Gamma_{1}, \cdots \Gamma_{m}over GF(q^{n}). A means of obtaining an encoderG(D)from the polynomial\Gamma(D)=\Gamma_{0}+\Gamma_{1}D+\cdots +\Gamma_{m}D^{m}with respect to a basis for GF(q^{n})over GF(q)is described. A criterion on\Gamma(D)in order for anyG(D)obtained from it to be noncatastrophic is established, which involves computing only the greatest common divisor (gcd) amongs=n/kpolynomials over GF(q^{k}). This criterion is shown to coincide with that of Massey and Sain whenk=1. It is shown that if\Gamma(D)is noncatastrophic (i.e., if encoders obtained from it are noncatastrophic) and has zero delay, then any encoderG(D)obtained from it is minimal and has a zero-delay feed-forward inverse. The number of zero-delay noncatastrophic polynomials over GF(q^{n})of degreemis shown to beq^{nm}(q^{n}-1)(q^{n-k}-1)/q^{n-k}(q^{k}-1), a formula which coincides with that of Shusta [11] whenk=1. The class of codes just described is shown to form a group under multiplication. If the basis is normal, the class is shown to be dosed under cyclic shifting. Whenk=1the class of codes described coincides with the class of all(n,1)F-ary convolutional codes; hence we obtain new proofs of certain well-known results about this latter class of codes. Finally, the binary rate1/2convolutional codes obtained from the noncatastrophic divisors ofD^{15}+1over GF(2^{2})are studied and optimal codes of constraint lengths6, 8, and12found.

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}.

A linear ensemble of codes is defined as one over which the informationK-tuple\proptois encoded as\proptoG \oplus_{z}whereGis equally likely to assume any matrix in a linear space\cal BofKbyNbinary matrices and wherezis independent ofGand equally likely to assume any binaryN-tuple. A technique for upperbounding the ensemble averageP(E)of the probability of error, when the codes of\cal Bare used on the binary symmetric channel with maximum likelihood decoding, is presented which reduces to overbounding a deterministic integer-valued function defined on the space of binaryN-tuples. This technique is applied to the ensemble of K by N binary matrices having for/th row the (i- 1) right cyclic shift of the first, i= 1,2,. . . ,K, and where the first row is equally likely to he any binaryN-tuple. For this ensemble it is shown thatP(E) \leq \mu(N) \exp_{2}-NE_{r}(K/N)whereE_{r}( \cdot)is the random coding exponent for the binary symmetric channel and_{ \mu}(N)is the number of divisors ofX^{N}+ 1. If\cal Bis pairwise independent it is shown that the above technique yields the random coding bound for block codes and that moreover there exists at least one code in the ensemble\cal Bwhose minimum Hamming distance meets a Gilbert-type lower bound.

Without recourse to the normal basis theorem, some(mk, k)cyclic codes are put into quasi-cyclic form, and their weight distributions are obtained. As a special case, some(3p,p)codes of Karlin are examined. It is shown that the dual(3p,2p)codes have minimum distance of at most six.

In this correspondence a method is presented whereby the average synchronization-error-correcting capability of Tavares' subset codes may be improved with no additional cost in rate and with only a small increase in the complexity of encoding and decoding. The method consists simply in shifting every word of the subset codes in such a way so that the shifted versions have a maximum number of leading and trailing zeros. A lower bound on the increase in synchronization-error-correcting capability provided by this method is derived.

Letg(x)h(x) = x^n - 1, n = q^m - 1, and assume thath(x)contains a primitive factorf(x)of degreem. IfVis theq-ary(n, k)cyclic code generated byg(x), Uits subcode generated byg(x)f(x), then it will be shown that the weight distribution ofVcan be obtained from the weight distribution ofUand its cosetU + g(x).