On cyclic MDS codes of length q over GF(q) (Corresp.)

It is shown that a cyclic codeCof lengthqover GF(q)is the maximum distance separable if and only if either1) qis a prime, in which caseCis equivalent, up to a coordinate permutation, to an extended Reed-Solomon code, or2) Cis a trivial code of dimensionk in {1, q - 1, q }. Hence there exists a nontrivial cyclic extended Reed-Solomon code of lengthqover GF(q)if and only ifqis a prime.     

An upper bound for codes for the noisy two-access binary adder channel (Corresp.)

Using earlier methods a combinatorial upper bound is derived for|C|. cdot |D|, where(C,D)is adelta-decodable code pair for the noisy two-access binary adder channel. Asymptotically, this bound reduces toR_{1}=R_{2} leq frac{3}{2} + elog_{2} e - (frac{1}{2} + e) log_{2} (1 + 2e)= frac{1}{2} - e + H(frac{1}{2} - e) - frac{1}{2}H(2e),wheree = lfloor (delta - 1)/2 rfloor /n, n rightarrow inftyandR_{1}resp.R_{2}is the rate of the codeCresp.D.     

Optimal source codes for geometrically distributed integer alphabets (Corresp.)

LetP(i)= (1 - theta)theta^ibe a probability assignment on the set of nonnegative integers wherethetais an arbitrary real number,0 < theta < 1. We show that an optimal binary source code for this probability assignment is constructed as follows. Letlbe the integer satisfyingtheta^l + theta^{l+1} leq 1 < theta^l + theta^{l-1}and represent each nonnegative integeriasi = lj + rwhenj = lfloor i/l rfloor, the integer part ofi/l, andr = [i] mod l. Encodejby a unary code (i.e.,jzeros followed by a single one), and encoderby a Huffman code, using codewords of lengthlfloor log_2 l rfloor, forr < 2^{lfloor log l+1 rfloor} - l, and lengthlfloor log_2 l rfloor + 1otherwise. An optimal code for the nonnegative integers is the concatenation of those two codes.     

On certain projective geometry codes (Corresp.)

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, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{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 numbersnu, such thatalpha^{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.     

On generator matrices of MDS codes (Corresp.)

It is shown that the family ofq-ary generalized Reed-Solomon codes is identical to the family ofq-ary linear codes generated by matrices of the form[I|A], whereIis the identity matrix, andAis a generalized Cauchy matrix. Using Cauchy matrices, a construction is shown of maximal triangular arrays over GF(q), which are constant along diagonals in a Hankel matrix fashion, and with the property that every square subarray is a nonsingular matrix. By taking rectangular subarrays of the described triangles, it is possible to construct generator matrices[I|A]of maximum distance separable codes, whereAis a Hankel matrix. The parameters of the codes are(n,k,d), for1 leq n leq q+ 1, 1 leq k leq n, andd=n-k+1.     

A note on maximum distance separable (optimal) codes (Corresp.)

Given a positive integerkand a prime powerqwithq leq k, it is proved that the largest value ofnfor which there exists an[n,k,n-k+l]maximum distance separable (MDS) code over GF(q)isk+1. A simple proof for the largest value ofnfor which there exists an[n,2,n-1]MDS code over any finite field is also given.     

Algorithms for the generation of full-length shift- register sequences

Two algorithms are presented for the generation of full-length shift-register cycles, also referred to as de Bruijn sequences. The first algorithm generates2^{k cdot g(n,k)full cycles of length2^{n}, using3n + k cdot g(n, k)bits of storage, wherekis a free parameter in the range1 leq k leq 2^{((n-4)/2)}, andg(n, k)is of the order ofn - 2 log k. The second algorithm generates about2^{n^{2}/4}full cycles of length2^{n}, using aboutn^{2}/2bits of storage. In both algorithms, the time required to produce the next bit from the lastnbits is close ton. A possible application to the construction of stream ciphers is indicated.     

On the survival of sequence information in filters (Corresp.)

Given a binary data streamA = {a_i}_{i=o}^inftyand a filterFwhose output at timenisf_n = sum_{i=0}^{n} a_i beta^{n-i}for some complexbeta neq 0, there are at most2^{n +1)distinct values off_n. These values are the sums of the subsets of{1,beta,beta^2,cdots,beta^n}. It is shown that all2^{n+1}sums are distinct unlessbetais a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space{f_n}is2^{n+1}ifbetais transcendental, ifbeta neq pm 1is rational, and ifbetais irrational algebraic but not a unit of the type mentioned. For the exceptional values ofbeta, it appears that the size of the state space{f_n}grows only as a polynomial innifmidbetamid = 1, but as an exponentialalpha^nwith1 < alpha < 2ifmidbetamid neq 1.     

A product construction for perfect codes over arbitrary alphabets (Corresp.)

A general product construction for perfect single-error-correcting codes over an arbitrary alphabet is presented. Given perfect single-error-correcting codes of lengthsn, m, andq + 1over an alphabet of orderq, one can construct perfect single-error-correcting codes of length(q - 1)nm + n + mover the same alphabet. Moreover, if there exists a perfect single-error-correcting code of lengthq + 1over an alphabet of orderq, then there exist perfect single-error-correcting codes of lengthn,n = (q^{t} _ 1)/(q - 1), and(t > 0, an integer). Finally, connections between projective planes of orderqand perfect codes of lengthq + 1over an alphabet of orderqare discussed.     

On the Probability of Undetected Error for the Maximum Distance Separable Codes

In this paper we investigate the performance of maximum-distance-separable codes with symbols fromGF(q)when they are used for pure error detection or for simultaneous error correction and detection over aq-input andq-output discret memoryless channel with symbol error probability ε. First we show that the probability of undetected error for an MDS code used for pure error detection is upper bounded byq^{-r}and decreases monotonically as εdecreases from(q - 1)/qto 0, whereris the number of parity-check symbols of the code. Then we show that the probability of undetected error for an MDS code used for correctingtor fewer symbol errors is upper bounded byq^{-r} Summin{i=0}max{t}(min{i} max{n})(q - 1)^{i}and decreases monotonically as ε decreases from(q - 1)/qto 0. These results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.     

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

It is shown thatsqrt[8]{2}is an element of order2^{n+4}inGF(F_{n}), whereF_{n}=2^{2^{n}}+1is a Fermat prime forn=3,4. Hence it can be used to define a fast Fourier transform (FFT) of as many as2^{n+4}symbols inGF(F_{n}). Sincesqrt[8]{2}is a root of unity of order2^{n+4}inGF(F_{n}), this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the casen=2, wheresqrt{2}is inGF(F_{2})=GF(17), is given.     

Improvements of Winograd's result on computation in the presence of noise (Corresp.)

Winograd's result concerning Elias' model of computation in the presence of noise can be stated without reference to computation. If a codevarphi: {0,1}^{k} rightarrow {0,1}^{n}is min-preserving(varphi (a wedge b) = varphi (a) wedge varphi (b)fora,b in {0,1}^{k})andepsilon n-error correcting, then the ratek/n rightarrow 0ask rightarrow infty. This result is improved and extended in two directions. begin{enumerate} item For min-preserving codes with {em fixed} maximal (and also average) error probability on a binary symmetric channel againk/n rightarrow 0ask rightarrow infty(strong converses). item Second, codes with lattice properties without reference to computing are studied for their own sake. Already for monotone codes( varphi (a) leq varphi (b)fora leq b)the results in direction 1) hold for maximal errors. end{enumerate} These results provide examples of coding theorems in which entropy plays no role, and they can be reconsidered from the viewpoint of multiuser information theory.     

Comma-free error-correcting codes

A method is shown by which it is possible to establish the existence (or nonexistence) of the comma-free properties of any group code from a simple observation of its null-space. Using this technique it is then demonstrated that all(n, k){em cyclic} group error-correcting code dictionaries can be made comma-free (without adding further redundancy or altering their error-correcting properties) if, and only if,k leq (n - 1)/2. Ifkis larger than(n - 1)/2it is still possible to detect bit gains or losses of up ton-k-1code bits.     

Writing on dirty paper (Corresp.)

A channel with outputY = X + S + Zis examined, The stateS sim N(0, QI)and the noiseZ sim N(0, NI)are multivariate Gaussian random variables (Iis the identity matrix.). The inputX in R^{n}satisfies the power constraint(l/n) sum_{i=1}^{n}X_{i}^{2} leq P. IfSis unknown to both transmitter and receiver then the capacity isfrac{1}{2} ln (1 + P/( N + Q))nats per channel use. However, if the stateSis known to the encoder, the capacity is shown to beC^{ast} =frac{1}{2} ln (1 + P/N), independent ofQ. This is also the capacity of a standard Gaussian channel with signal-to-noise power ratioP/N. Therefore, the stateSdoes not affect the capacity of the channel, even thoughSis unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the stateSrather than attempting to cancel it.     

A binary extremal doubly even self-dual code (64,32,12) obtained from an extended Reed-Solomon code over Fl6 (Corresp.)

The binary image of an extended Reed-Solomon code overF_{16}relative to a trace-orthogonal basis ofF_{l6}is an extremal doubly even self-dual code(64,32,12).     

Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We havefrac{1}{2} ln left( 1 + rho left( 1+ sum_{k=1}^{m} alpha_{k} x^{- k} right)^{2} right) leq C_{L} leq frac{1}{2} ln left( 1+ rho left( 1+ sum_{k = 1}^{m} alpha_{k} / sqrt{1 + rho} right)^{2} right), mbox{all rho}, whererho = P/N_{0}W, alpha_{l}, cdots, alpha_{m}are regression coefficients,Pis power,Wis bandwidth,N_{0}is the one-sided innovation spectrum, andxis a root of the polynomial(X^{2} - 1)x^{2m} - rho left( x^{m} + sum^{m}_{k=1} alpha_{k} x^{m - k} right)^{2} = 0.It is conjectured that the lower bound is the feedback capacity.     

High-radix transforms for Reed-Solomon codes over Fermat primes (Corresp.)

It is shown that a high-radix fast Fourier transform (FFT) with generatorgamma = 3over GF(F_{n}), whereF_{n} = 2^{2}^{n'} + 1is a Fermat prime, can be used for encoding and decoding of Reed-Solomon (RS) codes of length2^{2}^{n}. Such an RS decoder is considerably faster than a decoder using the usual radix 2 FFT. This technique applies most ideally to a 16-error-correcting, 256-symbol RS code of 8 bits being considered currently for space communication applications. This special code can be encoded and decoded rapidly using a high-radix FFT algorithm over GF(F_{3}).     

Using codes for error correction and detection (Corresp.)

A linear codeCover GF(q)is good fort-error-correction and error detection ifP(C,t;epsilon) leq P(C,t;(q - 1)/q)for allepsilon, 0 leq epsilon leq (q - 1)/q, whereP(C, t; epsilon)is the probability of an undetected error after a codeword inCis transmitted over aq-ary symmetric channel with error probabilityepsilonand correction is performed for all error patterns withtor fewer errors. A sufficient condition for a code to be good is derived. This sufficient condition is easy to check, and examples to illustrate the method are given.     

Coding the vertexes of a graph

Given a graphGofnnodes. We wish to assign to each nodei(i = 1, 2, cdots n)a unique binary codec_{i}of lengthmsuch that, if we denote the Hannuing distance betweenc_{i}andc_{j}asH(c_{i}, c_{j}), thenH(c_{i}, c_{j})leq Tif nodesiandjare adjacent (i.e., connected by a single branch), andH(c_{i}, c_{j}) geq T+1otherwise. If such a code exists, then we say thatGis doable for the value ofTand tn associated with this code. In this paper we prove various properties relevent to these codes. In particular we prove 1) that for every graphGthere exists anmandTsuch thatGis doable, 2) for every value ofTthere exists a graphG̃which is notTdoable, 3) ifGisT'doable, then it isT'+ 2pdoable forp = 0, 1, 2, cdots, and is doable for allT geq 2T'ifT'is odd, and is doable for allT geq 2T' + 1ifT'is even. In theory, the code can be synthesized by employing integer linear programming where eitherTand/ormcan be minimized; however, this procedure is computationally infeasible for values ofnandmin the range of about10or greater.     

Multiple-burst error-correcting cyclic product codes (Corresp.)

LetCbe the cyclic product code ofpsingle parity check codes of relatively prime lengthsn_{1}, n_{2},cdots , n_{p} (n_{1} < n_{2} < cdots < n_{p}). It is proven thatCcan correct2^{P-2}+2^{p-3}-1bursts of lengthn_{1}, andlfloor(max{p+1, min{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}}}-1)/2rfloorbursts of lengthn_{1}n_{2} cdots n_{s} (2leq s leq p-2). Forp=3this means thatCis double-burst-n_{1}-correcting. An efficient decoding algorithm is presented for this code.