On the undetected error probability of nonlinear binary constantweight codes

The undetected error probability (UEP) of binary (n, 2δ, m) nonlinear constant weight codes over the binary symmetric channel (BSC) is investigated, where n is the blocklength, m is the weight of codeword and 2δ is the minimum distance of the codes. The distance distribution of the (n, 2, m) nonlinear constant weight codes is evaluated. It is proven in this paper that the (5, 2, 2) code, (5, 2, 3) code, (6, 2, 3) code, (7, 2, 4) code, (7, 2, 3) code and (8, 2, 4) code are the only proper error-detecting codes in the (n, 2, m) nonlinear constant weight codes for n⩾5, in the sense that their UEP is increased monotonically with the channel error rate p, of course all these proper codes are m-out-of-n codes. Furthermore, it is conjectured that except for the cases of n⩽4δ, there are no proper error-detecting binary (n, 2δ, m) nonlinear constant weight codes, for n>8 and δ⩾1     

On some properties of the undetected error probability of linear codes (Corresp.)

A recent paper [1] discussed the2^{-p}bound (wherep = n- k) for the probability of undetected errorP(epsilon)for an(n,k)block code used for error detection on a binary symmetric channel. This investigation is continued and extended and dual codes are studied. The dual and extension of a perfect code obey the2^{-p}bound, but this is not necessarily true for arbitrary codes that obey the bound. Double-error-correcting BCH codes are shown to obey the bound. Finally the problem of constructing uniformly good codes is examined.     

A lower bound on the undetected error probability and strictlyoptimal codes

Error detection is a simple technique used in various communication and memory systems to enhance reliability. We study the probability that a q-ary (linear or nonlinear) block code of length n and size M fails to detect an error. A lower bound on this undetected error probability is derived in terms of q, n, and M. The new bound improves upon other bounds mentioned in the literature, even those that hold only for linear codes. Block codes whose undetected error probability equals the new lower bound are investigated. We call these codes strictly optimal codes and give a combinatorial characterization of them. We also present necessary and sufficient conditions for their existence. In particular, we find all values of n and M for which strictly optimal binary codes exist, and determine the structure of all of them. For example, we construct strictly optimal binary-coded decimal codes of length four and five, and we show that these are the only possible lengths of such codes     

The worst case probability of undetected error for linear codes onthe local binomial channel

The worst case probability of undetected error for a linear [n,k:q] code used on a local binomial channel is studied. For the two most important cases it is determined in terms of the weight hierarchy of the code. The worst case probability of undetected error is determined explicitly for some classes of codes     

Special hardware for computing the probability of undetected errorfor certain binary CRC codes and test results

A hardware device for efficiently evaluating the probability of undetected error for a class of CRC error detection codes with a large number of parity check digits is described. The generator polynomial for the codes in this class are of the form g(x)=(1+x)p(x) where p(x) is a primitive irreducible polynomial. The degree of g(x), R, is the number of parity check digits. Using this hardware, a search was conducted for codes in this class (for 8⩽R⩽39) which are “proper” for shortened block lengths. A table of codes satisfying this condition is included     

Distance distribution of binary codes and the error probability of decoding

We address the problem of bounding below the probability of error under maximum-likelihood decoding of a binary code with a known distance distribution used on a binary-symmetric channel (BSC). An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the "random coding exponent" is the true value of the channel reliability for codes rate R in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.     

Computing the probability of undetected error for shortened cycliccodes

The authors present a general technique for computing P _e for all possible shortened versions of cyclic codes generated by any given polynomial. The technique is recursive, i.e. computes P_e for a given code block length n from that of the code block length n-1. The proposed computation technique for determining P_e does not require knowledge of the code weight distributions. For a generator polynomial of degree r, and |g| nonzero coefficients, the technique yields P_e for all code block lengths up to length n in time complexity O(n|g |2^r+|g|). Channels with variable bit error probabilities can be analyzed with the same complexity. This enables the performance of the code generator polynomials to be analyzed for burst errors     

On the decoder error probability of block codes

By using coding and combinational techniques, an explicit formula is derived which enumerates the complete weight distribution of decodable words of block codes using partially known weight distributions. Also, an approximation formula for nonbinary block codes is obtained. These results give exact and approximate expressions for the decoder error probability P_E(u) of block codes     

Bounds on the decoding error probability of binary linear codes viatheir spectra

Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it     

A union bound on the error probability of binary codes over block-fading channels

Block-fading is a popular channel model that approximates the behavior of different wireless communication systems. In this paper, a union bound on the error probability of binary-coded systems over block-fading channels is proposed. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel. The distribution of error bits over the fading blocks is computed. For a specific distribution pattern, the pairwise error probability is derived. Block-fading channels modeled as Rician and Nakagami distributions are studied. We consider coherent receivers with perfect and imperfect channel side information (SI) as well as noncoherent receivers employing square-law combining. Throughout the paper, imperfect SI is obtained using pilot-aided estimation. A lower bound on the performance of iterative receivers that perform joint decoding and channel estimation is obtained assuming the receiver knows the correct data and uses them as pilots. From this, the tradeoff between channel diversity and channel estimation is investigated and the optimal channel memory is approximated analytically. Furthermore, the optimal energy allocation for pilot signals is found for different channel memory lengths.     

Bounds on the worst case probability of undetected error

Upper and lower bounds on the average worst-case probability of undetected error for linear [n,k,q] codes are given     

More on the decoder error probability for Reed-Solomon codes

A combinatorial technique similar to the principle of inclusion and exclusion is used to obtain an exact formula for P_E (u), the decoder error probability for Reed-Solomon codes. The P_E(u) for the (255, 223) Reed-Solomon code used by NASA and for the (31, 15) Reed-Solomon code (JTIDS code) are calculated using the exact formula and are observed to approach the Qs of the codes rapidly as u gets large. An upper bound for the expression |P_E(u)/ Q-1| is derived and shown to decrease nearly exponentially as u increases     

On the decoder error probability for Reed - Solomon codes (Corresp.)

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for aterror-correcting Reed-Solomon code of lengthq - 1over GF(q), if more thanterrors occur, the probability of decoder error is less than1/t!.     

Bounds on the undetected error probabilities of linear codes forboth error correction and detection

The author investigates the (n, k, d⩾2t+1) binary linear codes, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel. In particular, for t=1, it is shown that there exists one code whose probability of undetected errors is upper-bounded by (n+1) [2^n-k-n]^-1 when used on a binary symmetric channel with transition probability less than 2/n     

A simple derivation of the undetected error probabilities of complementary codes

Recently, Fu, Klove, and Wei (2003) have shown that the undetected error probability of a binary code is related to that of its complement, and the undetected error probability of a constant-weight binary code is related to that of its complement relative to the set of all constant-weight vectors. We generalize these relations to cover the complements of any binary or nonbinary code relative to a distance-invariant code containing the first code. We prove the generalization using a much simpler argument than the published proofs of the special cases.     

On the probability of error of antenna-subset selection with space-time block codes

Orthogonal space-time block codes (OSTBCs) can obtain full diversity advantage with a simple, but optimal, receiver. Unfortunately, OSTBCs lack in array gain compared with beamforming techniques and suffer a rate loss for more than two transmit antennas. One simple method for improving the array gain and adapting OSTBCs to any number of transmit antennas is antenna-subset selection, where the OSTBC is transmitted on a subset of the transmit antennas. In this letter, we analyze the symbol-error rate performance of antenna-subset selection combined with OSTBCs.     

On zero error probability of binary decisions (Corresp.)

In view of a certain randomized decision scheme, we simplify Rényi's result that concerns the conditions that a changing situation for binary decision must satisfy in order that the probability of decision errors tend to zero. We give its possible interpretation for a communication channel that has a tendency to become useless with time.     

Concerning a bound on undetected error probability (Corresp.)

In the past, it has generally been assumed that the probability of undetected error for an(n,k)block code, used solely for error detection on a binary symmetric channel, is upperbounded by2^{-(n-k)}. In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.     

The simplex codes and other even-weight binary linear codes for error correction

The probability of correct decoding on the binary-symmetric channel is studied. In particular, a class of codes with the same lengths and dimensions as the linear simplex codes, but with larger probability of correct decoding for all parameters p, 0 < p < 1/2, is given.     

Exact pairwise error probability of space-time codes

We describe a simple technique for the numerical calculation, within any desired degree of accuracy, of the pairwise error probability (PEP) of space-time codes over fading channels. This method applies also to the calculation of E[Q(√ξ)] for any nonnegative random variable ξ whose moment-generating function Φ_ξ(s)=E[exp(-sξ)] is known. Its application to the multiple antenna independent Rayleigh-fading channel and to the Rayleigh block fading channel is discussed, and illustrated by two simple examples