Sliding block codes between constrained systems

The construction of finite-state codes between constrained systems called sofic systems introduced by R. Karabed and B. Marcus (1988) is continued. It is shown that if Σ is a shift of finite type and S is a sofic system with k/n=h(S )/h(Σ), where h denotes entropy, there is a noncatastrophic finite-state invertible code from Σ to S at rate k:n if Σ and S satisfy a certain algebraic condition involving dimension groups, and Σ and S satisfy a certain condition on their periodic points. Moreover, if S is an almost finite type sofic system, then the decoder can be sliding block     

Short codes with a given covering radius

The covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword. The authors introduce a new function l(m,r), called the length function, which equals the smallest length of a binary code of codimension m and covering radius r. They investigate basic properties of the length function. Projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l(m,2) obtained by amalgamation of Hamming codes. General methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l(m,2). A table is presented which gives the best results now known for l(m,r) with m⩽12 and r⩽12     

Limited search trellis decoding of convolutional codes

The least storage and node computation required by a breadth-first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated. Breadth-first decoders work with code paths of the same length, without backtracking. The Viterbi algorithm is an exhaustive trellis decoder of this type; other schemes look at a subset of the tree or trellis paths. For random tree codes, theorems about the asymptotic number of paths required and their depth are proved. For concrete convolutional codes, the worst case storage for t error sequences is measured. In both cases the optimal decoder storage has the same simple dependence on t. The M algorithm and algorithms proposed by G.J. Foschini (ibid., vol.IT-23, p.605-9, Sept. 1977) and by S.J. Simmons (PhD. diss., Queens Univ., Kingston, Ont., Canada) are optimal, or nearly so; they are all far more efficient than the Viterbi algorithm     

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     

Syndrome and transition count are uncorrelated [logic circuittesting]

In the testing of logic circuits, two proposed data-compression methods use the number of ones (syndrome) and the number of sequence changes (transition count). An enumeration N(m, k , t) of the number of length-m binary sequences having syndrome value k and transition count t is developed. Examination of this result reveals that the parallel compression of these two methods has small overlap in error masking. An asymptotic expression for N(m, k, t) is developed     

Bounds and constructions for codes correcting unidirectional errors

A brief introduction is given on the theory of codes correcting unidirectional errors, in the context of symmetric and asymmetric error-correcting codes. Upper bounds on the size of a code of length n correcting t or fewer unidirectional errors are then derived. Methods in which codes correcting up to t unidirectional errors are constructed by expurgating t-fold asymmetric error-correcting codes or by expurgating and puncturing t -fold symmetric error-correcting codes are also presented. Finally, tables summarizing some results on the size of optimal unidirectional error-correcting codes which follow from these bounds and constructions are given     

Upper bounds for small trellis codes

An upper bound on the minimum squared distance of trellis codes by packing Voronoi cells is derived and compared with previously known bounds. The authors focus on codes with small memory for modulation formats such as pulse amplitude modulation (PAM), m-ary quadrature amplitude modulation (QAM), and m-ary phase shift keying (PSK). The bound is tight to search results for coset codes with a small number of states     

Energy consumption in multilective and boundary VLSI computations

A VLSI computation is said to be m-way multilective when each input bit is available m times in either space or time or both. The repeated availability of input bits can save computational energy. For a uniswitch m-way multilective computation, where a wire can switch at most once, it is shown that the energy savings can be as much as a factor of m. A multiswitch m-way multilective computation can save up to a factor of √ m switching energy. Tighter energy lower bounds are derived for a circuit with the input/output (I/O) pads located on the border. These boundary computations seem to cost an additional factor ranging from √log n to log n in switching energy. The author extends the energy lower bounds for the multilective case, for a chip with aspect ratio a. The additional energy cost ranges from a factor of √a to a factor of a     

Binary [18,11]2 codes do not exist-Nor do [64,53]2 codes

A binary, linear block code C with block length n and dimension n is commonly denoted by [n, k] or, if its minimum distance is d, by [n, k,d]. The code's covering radius r(C) can be defined as the smallest number r such that any binary column vector of length (n-k) can be written as a sum of r or fewer columns of a parity-check matrix of C. An [n,k] code with covering radius r is denoted by [n,k]r. R.A. Brualdi et al., (1989) showed that l(m,r) is defined to be the smallest n such that an [n,n-m]r code exists. l(m,2) is known for m⩽6, while it is shown by Brualdi et al. that 17⩽l(7,2)⩽19. This lower bound is improved by A.R. Calderbank et al. (1988), where it is shown that [17,10]2 codes do not exist. The nonexistence of [18,11]2 codes is proved, so that l(7,2)=19. l[7.2)=19 is established by showing that [18,11]2 codes do not exist. It is also shown that [64,53]2 codes do not exist, implying that l(11,2)⩾65     

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     

Construction of t-EC/AUED codes

A t-EC/AUED code is constructed by appending a single check symbol from an alphabet S to each word of an n-bit binary t-EC code of even weight. Conditions are derived for the construction of S and a procedure is given which, for some values of t, n, leads to codes with fewer check bits than known codes with equivalent properties     

Pseudocyclic maximum-distance-separable codes

The (n, k) pseudocyclic maximum-distance-separable (MDS) codes modulo (xⁿ- a) over GF(q) are considered. Suppose that n is a divisor of q+1. If n is odd, pseudocyclic MDS codes exist for all k. However, if n is even, nontrivial pseudocyclic MDS codes exist for odd k (but not for even k) if a is a quadratic residue in GF(q), and they exist for even k (but not for odd k) if a is not a quadratic residue in GF(q). Also considered is the case when n is a divisor of q-1, and it is shown that pseudocyclic MDS codes exist if and only if the multiplicative order of a divides (q-1)/n, and that when this condition is satisfied, such codes exist for all k. If the condition is not satisfied, every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code     

Sliding-block coding for input-restricted channels

Work on coding arbitrary sequences into a constrained system of sequences (called a sofic system) is presented. Such systems model the input constraints for input-restricted channels (e.g., run-length limits and spectral constraints for the magnetic recording channel). In this context it is important that the code be noncatastrophic to ensure that the decoder has limited error propagation. A constructive proof is given of the existence of finite-state invertible noncatastrophic codes from arbitrary n-ary sequences to a sofic system S at constant rate p:q provided only that Shannon's condition (p/q)⩽(h/log n) is satisfied, where h is the entropy of the system S. If strict inequality holds or if equality holds and S satisfies a natural condition called `almost of finite type' (which includes the systems used in practice), a stronger result is obtained, namely, the decoders can be made `state-independent' sliding-block. This generalizes previous results. An example is also given to show that the stronger result does not hold for general sofic systems     

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     

Minimal distance lexicographic codes over an infinite alphabet

The author investigates the properties of minimal distance lexicographic codes, or lexicode, over the ordered infinite alphabet N={0,1,2…}. The author presents a method for computing the basis of such a code. It is shown that any lexicographic code S with minimal distance d has a unique basis where each basis vector is a one followed by a string of zeros, followed by d-1 nonzero digits a_ij. Furthermore, the matrix A=(a_ij) has no singular minors over the nim-field. The dual code when S has finite length is also computed. The author develops a systematic approach to determine which words belong to these lexicodes     

New lower bounds for covering codes

Some new lower bounds on |C| for a binary linear [n, k]R code C with n+1=t(R +1)-r(0⩽r<R+1, t>2 odd) or with n+1=t(R+1)-1(t>2 even) are obtained. These bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t[n, k], the smallest covering radius of any [n, k] code     

Applicability of physical optics thin plate scattering formulas forremote sensing

The authors evaluate the applicability of simple formulas for scattering from flat plates that were developed using an extended physical optics (PO) procedure. These formulas take on especially simple forms (denote here as TPO) when the plates are electrically very thin. The authors consider circular plates (disks) and show that when the radius-to-thickness ratio (a/t) is large, the TOP formulas give accurate backscatter cross sections for all incident angles. The PO formulas are not usable for angles of incidence near or at edge-on to the flat surfaces of the disk. On the other hand, complex polarization rate information is lost with TPO. It has been argued elsewhere that TOP should hold even for electrically small disks provided a/t is large. The authors show that TPO gives results accurate (with some exceptions) to ~4% when a/t ~200 for Rayleigh disks. These results are obtained primarily by comparing TPO computational results with an exact numerical procedure     

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     

Decision depths of convolutional codes

The decision depth function of a convolutional code L_D(w) is the first trellis depth at which all unmerged incorrect subset paths exceed weight w. The authors tabulate this function for most codes in common use. L _D(w) specifies the path memory length of a decoder. It follows a clear asymptotic growth rule     

Orthogonality of binary codes derived from Reed-Solomon codes

The author provides a simple method for determining the orthogonality of binary codes derived from Reed-Solomon codes and other cyclic codes of length 2^m-1 over GF(2^m) for m bits. Depending on the spectra of the codes, it is sufficient to test a small number of single-frequency pairs for orthogonality, and a pair of bases may be tested in each case simply by summing the appropriate powers of elements of the dual bases. This simple test can be used to find self-orthogonal codes. For even values of m, the author presents a technique that can be used to choose a basis that produces a self-orthogonal, doubly-even code in certain cases, particularly when m is highly composite. If m is a power of 2, this technique can be used to find self-dual bases for GF(2 ^m). Although the primary emphasis is on testing for self orthogonality, the fundamental theorems presented apply also to the orthogonality of two different codes