Minimum Euclidean distance for combinations of short rate 1/2 convolutional codes and CPFSK modulation

Continuous phase frequency shift keying (CPFSK) is a constant amplitude modulation method with good spectral sidelobe properties. Good error probability properties can be obtained with coherent maximum-likelihood detection. In this paper we study the Euclidean distance properties of signals formed by a conventional rate1/2convolutional encoder followed by a binary or4-level CPFSK modulator. The minimum Euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length. The optimum detector is discussed for rational modulation index values. The best obtainable codes are found for the case of short rate1/2codes with binary or4-level CPFSK modulation. Lists of the best codes are given. Among the results are that the noncatastrophic rate1/2convolutional codes with optimum free Hamming distance do not in general give the best Euclidean distance with CPFSK.     

Odd weight symmetry in some binary codes (Corresp.)

If a linear binary code of lengthncontains the all-one codeword, than the weights of the code are symmetric. We consider those codes which do not contain the all-one codeword and yet have an equal number of symmetrically placed odd weight words.     

On the minimum distance of some BCH codes (Corresp.)

Some new examples of binary Bose-Chaudhuri-Hocquanghen (BCH) codes of length 255 are found for which the minimum distance and designed distance agree.     

Equidistant binary arithmetic codes (Corresp.)

LetC(B)denote the binary cyclicANcode with generatorA, whereAB=2^{n} - 1. It is known thatC(B)is equidistant ifBis a prime powerp^{k}, where either2or-2is primitive moduloBprovidedpequiv 1 pmod{3}{rm if} k > 1. It is conjectured that these are the onlyBsuch thatC(B)is equidistant. We have verified this forB < 100 000. Several results are established that further limit the possibilities for counterexamples to the conjecture.     

On simplex-type binary codes (Corresp.)

The rank of the generating matrix in certain cases is computed. The result is perhaps of interest in its own right, apart from any applications it may have. The method of proof, utilizing as it does a knowledge of the parameters of a certain association scheme, is a technique which can be profitably used elsewhere.     

Weight distributions of some classes of binary cyclic codes (Corresp.)

Leth_1(x)h_2(x)be the parity-check polynomial of a binary cyclic code. This correspondence presents a formula for decomposing words in the code as sums of multiples of words in the codes whose parity-check polynomials areh_1(x)andh_2(x). This decomposition provides information about the weight distribution of the code.     

On the minimum distance of some quadratic residue codes (Corresp.)

We find the minimum distances of the binary(113, 57), and ternary(37, 19), (61, 31), (71, 36), and(73, 37)quadratic residue codes and the corresponding extended codes. These distances are15, 10, 11, 17, and17, respectively, for the nonextended codes and are increased by one for the respective extended codes. We also characterize the minimum weight codewords for the(113, 57)binary code and its extended counterpart.     

Construction for binary asymmetric error-correcting codes (Corresp.)

A construction method of asymmetric error-correcting codes is proposed. In this method, at-fold asymmetric error-correcting code of lengthn-1is constructed by expurgating and puncturing anyt-fold symmetric error-correcting code of lengthn. These codes are designed for use on binary asymmetric channels, such as large-scale integration (LSI) memory protection, on which transition is one-way.     

Short unit-memory byte-oriented binary convolutional codes having maximal free distance (Corresp.)

It is shown that(n_{o},k_{o})convolutional codes with unit memory always achieve the largest free distance among all codes of the same ratek_{o}/n_{o}and same number2^{Mk_{o}of encoder states, whereMis the encoder memory. A unit-memory code with maximal free distance is given at each place where this free distance exceeds that of the best code withk_{o}andn_{o}relatively prime, for allMk_{o} leq 6and forR = l/2, 1/3, 1/4, 2/3. It is shown that the unit-memory codes are byte oriented in such a way as to be attractive for use in concatenated coding systems.     

Synchronization of binary source codes (Corresp.)

The problem of achieving synchronization for variable-length source codes is addressed through the use of self-synchronizing binary prefix-condition codes. Although our codes are suboptimal in the sense of minimum average codeword length, they have the advantages of being generated by an explicit constructive algorithm, having minimal additional redundancy compared with optimal codes-as little as one additional bit introduced into the least likely codeword for a large class of sources-and having statistical synchronizing performance that improves on that of the optimal code in many cases.     

On synchronizable binary cyclic codes (Corresp.)

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.     

On binary majority-logic decodable codes (Corresp.)

LetVprimebe a binary(n,k)majority-logic decodable code withgprime (X)as its generator polynomial and odd minimum distanced. LetVbe the(n, k - 1)subset code generated bygprime (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.     

Further results on binary convolutional codes with an optimum distance profile (Corresp.)

Fixed binary convolutional codes are considered which are simultaneously optimal or near-optimal according to three criteria: namely, distance profiled, free distanced_{ infty}, and minimum number of weightd_{infty}paths. It is shown how the optimum distance profile criterion can be used to limit the search for codes with a large value ofd_{infty}. We present extensive lists of such robustly optimal codes containing rateR = l/2nonsystematic codes, several withd_{infty}superior to that of any previously known code of the same rate and memory; rateR = 2/3systematic codes; and rateR = 2/3nonsystematic codes. As a counterpart to quick-look-in (QLI) codes which are not "transparent," we introduce rateR = 1/2easy-look-in-transparent (ELIT) codes with a feedforward inverse(1 + D,D). In general, ELIT codes haved_{infty}superior to that of QLI codes.     

An asymptotic Hamming bound on the feedback decoding minimum distance of linear tree codes (Corresp.)

An asymptotic Hamming bound for tree codes is derived. This bound is not new, it has been obtained earlier by Pinsker. However, Markov chains were used by Pinsker in its proof whereas more direct methods are used here.     

On optimal codes for binary asymmetric channels (Corresp.)

Let{ v_{1}, cdots , v_{N} }be a code ofNvectors whose components consist ofmones and(n - m)zeros. The vectors are transmitted with equal probability over the asymmetric channel whose error probabilities for transmitted zeros and ones equal respectively0andp, 0 < p < 1(such channels arise in optical communication). The problem of determining for fixed values of the parametersN, n, m, pthose codes which achieve minimum error probability is studied. The problem is solved forN leq 3. Partial results are obtained for allN.