Gérald E. Séguin

A lower bound on the error probability for signals in white Gaussian noise

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 Poltyrevs upper bound.

Structural properties and enumeration of quasi cyclic codes

Given any finite fieldFq, an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofFqN which is invariant underTnfor some integern, 0 <n ≦N, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theFq[λ]-submodules ofFqN where the productμ(gl)·ν is naturally defined asμ0ν+μ1νTn+...+μmνTmnifμ(λ)= μ0+μ1+...+μmλm.In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicFq[λ]-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.

A class of 1-generator quasi-cyclic codes

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.

The q-ary image of a q/sup m/-ary cyclic code

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. >

A Class of High Rate Codes for Byte-Oriented Information Systems

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.

The trace description of irreducible quasi-cyclic codes

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. >

Error correction/masking for digital voice transmission over the land mobile satellite system

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. >

Low complexity normal bases for F 2 mn

Abstract If C ( r ) denotes the minimum complexity of a normal basis for F 2 r , we show that if m > 1, n > 1 are two relatively prime integers, then F 2 mn has a normal basis of complexity C ( m ) C ( n ). Such a normal basis leads to a Massey—Omura multiplier for F 2 mn which uses C ( m ) C ( n ) XOR gates and C ( m ) C ( n ) + 1 AND gates per dimension.

A counter-example to a recent result on the q -ary image of a q s -ary cyclic code

We show, by means of a counter-example, that the necessary and sufficient conditions given in a recent paper [3] in order for theq-ary image of aqs-ary cyclic code to be cyclic are incorrect.

A random coding bound for fixed convolutional codes of rate 1/n

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) >