Relationships between m-sequences over GF(q) andGF(qm)

It is shown that m-sequences over GF(q^m ) of length q^nm-1 corresponding to primitive polynomials in GF[q^m,x] of degree n can be generated from known m-sequences over GF(q) of length q^nm-1 obtained from primitive polynomials in GF[q,x] of degree mn. A procedure for generating the m-sequences over GF(q²) from m-sequences over GF(q) was given which enables the generation of m-sequences over GF( p²ⁿ). In addition it was shown that all of the primitive polynomials in GF[q,^m,x] can be obtained from a complete set of the primitive polynomials in GF[q ,x]     

A construction of non-Reed-Solomon type MDS codes

A construction is presented of long maximum-distance-separable (MDS) codes that are not generalized Reed-Solomon (GRS) type. The construction uses subsets S,|S|=m of a finite field F=GF(q) with the property that no t distinct elements of S add up to some fixed element of F . Large subsets of this kind are used to construct [n=m+2, k=t+1] non-GRS MDS codes over F     

Self-reciprocal polynomials and generalized Fermat numbers

Self-reciprocal polynomials (SRPs) over GF(q), where q is a prime power, q=p^k, are investigated. The maximum possible component for these polynomials is found for q odd. The construction of Fermat maximum exponent self-reciprocal polynomials (MRPs) over GF(2) is extended to GF(2^k ) with the aid of generalized Fermat numbers. These polynomials leads to a bound on the maximum possible exponent of SRPs over GF(2^k), and a simplified algorithm for finding these MRPs. Self-reciprocal polynomials have applications in cryptography, error-correction coding, and the synthesis of linear feedback shift registers. They are advantageous when available memory or hardware is restricted or when data can be read in either direction. Some results on quasi-self-reciprocal polynomials are also presented     

On the existence of optimum cyclic burst correcting codes over GF(q)

A cyclic b-burst correcting code over GF(q) of redundancy r and length n=(q^r-b+1-1)/(q-1) is said to be optimum. It is proved that a necessary condition for the existence of such a code is the existence of a square-free polynomial in GF(q)[x] of degree b-1 which is not divisible by x such that its period and the degrees of its irreducible factors are relatively prime to q-1. Moreover, if such a polynomial exists, then there are an infinite number of optimum cyclic b-burst correcting codes over GF(q)     

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GFⁿ¹(q) (n1⩽q ), and it is shown how to construct both multilevel block codes and multilevel trellis codes over GF(q). Multilevel (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d_min⩾d are presented. These codes use Reed-Solomon codes as component codes. Longer multilevel block codes are also constructed using q-ary block codes with block length longer than q+1 as component codes. Some quaternary multilevel block codes are presented with the same length and number of information symbols as, but larger distance than, the best previously known quaternary one-level block codes. It is proved that if all the component block codes are linear. the multilevel block code is also linear. Low-rate q-ary convolutional codes, word-error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multilevel trellis codes over GF(q) or binary-to-q-ary trellis codes     

A generalization of consecutive-k-out-of-n:Fsystems

A linear (m, n)-lattice system consists of m ·n elements arranged like the elements of a (m ,n)-matrix, i.e. each of the m rows includes m elements, and each of the n columns includes m elements. A circular (m,n)-lattice system consists of m circles (centered at the same point) and n rays. The intersections of the circle and the rays represent the elements, i.e. each of the circles includes n elements and each of the rays has m elements. A (linear or circular) (m, n)-lattice system is a (linear or circular) connected-X-out-of-(m,n):F lattice system if it fails whenever at least one subset of connected failed components occurs which includes failed components connected in the meaning of connected-X. The paper presents some practical examples and the reliability formulas of simple systems using results of consecutive-k-out-of-n:F systems     

Focused codes for channels with skewed errors

Consider a channel with inputs and outputs in the field F _q(q>2). It is said that the channel is skewed on a set B⊂F_q* if the additive noise generated by the channel is likely to lie in B, i.e. B is a set of common errors. The concern is the construction of focused codes that are appropriate for such channels. It is said that a code is (t₁,t₂)-focused on B if it can correct up to t₁+t₂ errors provided at most t₁ of those errors lie outside of B; the strategy is to offer different levels of protection against common and uncommon errors and so provide novel tradeoffs between performance and rate. Techniques for constructing focused codes and bounds on their rates are described     

On the linear complexity of functions of periodic GF(q)sequences

It is proved that the product of arbitrary periodic GF(q) sequences attains maximum linear complexity if their periods are pairwise coprime. The necessary and sufficient conditions are derived for maximum linear complexity of the product of two periodic GF(q ) sequences with irreducible minimal characteristic polynomials. For a linear combination of products of arbitrary periodic GF(q) sequences, it is shown that maximum linear complexity is achieved if their periods are pairwise coprime and the polynomial x -1 does not divide any of their minimal characteristic polynomials; assuming only that their periods are pairwise coprime, the author establishes a lower bound on the linear complexity which is of the same order of magnitude as maximum linear complexity. Boolean functions are derived that are optimal with respect to the maximum linear complexity. Possible applications of the results in the design of sequence generators are discussed     

Repeated-root cyclic codes

In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and F_q is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known | u|u+v| construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given     

Asymptotic performance of M-ary orthogonal modulation ingeneralized fading channels

The asymptotic (M→∞) probability of symbol error P_e,_m for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E_b/N₀) and E_b is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E_b/N₀. These channels exist for all m⩾1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel     

Identities and approximations for the weight distribution of q-ary codes

An explicit formula is derived that enumerates the complete weight distribution of an (n, k, d) linear code using a partially known weight distribution. An approximation formula for the weight distribution of q-ary linear (n, k , d) codes is also derived. It is shown that, for a given q-ary linear (n, k, d) code, the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q=q ^-(n-k) as u becomes large. The error term is a decreasing function of the minimum weight of the dual. The results are also valid for nonlinear (n, M, d) codes with the minimum weight of the dual replaced by the dual distance     

Lower bounds for q-ary covering codes

The authors prove combinatorial lower bounds for K_q (n,R), the minimal cardinality of any q-ary code of length n and covering radius R. Tables of lower bounds for K_q(n,R) are presented for q=3, 4, 5     

1/f noise in MODFETs at low drain bias

The 1/f noise in normally-on MODFETs biased at low drain voltages is investigated. The experimentally observed relative noise in the drain current S_I/I² versus the effective gate voltage V_G=V_GS-V_off shows three regions which are explained. The observed dependencies are S_I/I²∝V_G ^m with the exponents m=-1, -3, 0 with increasing values of V_G. The model explains m =-1 as the region where the resistance and the 1/f noise stem from the 2-D electron gas under the gate electrode; the region with m=0 at large V_G or V_GS≅0 is due to the dominant contribution of the series resistance. In the region at intermediate V_G , m=-3, the 1/f noise stems from the channel under the gate electrode, and the drain-source resistance is already dominated by the series resistance     

Minimum distance of logarithmic and fractional partialm-sequences

Two results are presented concerning the partial periods (p-p's) of an m-sequence of period 2ⁿ-1. The first proves the existence of an m-sequence whose p-p's of length approximately (n+d log₂ n) have minimum distance between d and 2d for small d. The second result is of an asymptotic nature and proves that the normalized minimum distance of p-p's whose length is any fraction of the period of the m-sequence, approaches 1/2 as the period of m-sequence tends to infinity     

Distribution of the lifetime of consecutive k-within-m-out-of-n:F systems

The distribution of the lifetime (MTTF) of any consecutive k -within-m-out-of-n:F system, with independent exponentially distributed component lifetimes, is shown to be a convex combination of the distributions (MTTFs) of several convolutions of independent random variables, where each convolution represents a distinct path in the evolution of the system's history, and where in each convolution all but the last random variable is exponential. The last random variable in each convolution is either a zero lifetime or the lifetime of several disjoint consecutive k_i within m_i-out-of-n:F systems in series with each k_i<k, each m_i<m, and each n_i<n. This enables the calculations to proceed recursively. Calculations are facilitated by the symmetric nature of the convex combination     

Effective recursive algorithm for computing multiplicative inversesin GF(2m)

Presents an effective recursive algorithm for computing multiplicative inverses in GF(2^m), where m=2^k, employing normal bases. The proposed algorithm requires m-1 cyclic shifts and two multiplications in GF (2^m) and in each subfield of GF(2^m): GF(2^m/2), GF(2^m/4),. . ., GF (2⁸) and GF(2⁴)     

m-consecutive-k-out-of-n:F systems

An m-consecutive-k-out-of-n:F system, consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Three theorems concerning such systems are stated and proved. Theorem one is a recursive formula to compute the failure probability of such a system. Theorem two is an exact formula for the failure probability. Theorem three is a limit theorem for the failure probability     

Fast decoding of codes from algebraic plane curves

Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m² /8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding the error locator is O(n^7/3 ), where n is the length of the code. For codes from Hermitian curves the complexity of finding the error values, given the error locator, is O(n²), and the same complexity can be obtained in the general case if only d*/2-m²/2 errors are corrected     

A spectral characterization of correlation-immune combiningfunctions

It is shown that a Boolean combining function f(x) of n variables is mth-order correlation-immune if and only if its Walsh transform F(ω) vanishes for all ω with Hamming weight between 1 and m, inclusive. This result is used to extend slightly Siegenthaler's (IEEE Trans. Comput., vol. C-34, pp. 81-85, Jan. 1985) characterization of the algebraic normal form of correlation-immune combining functions     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used