Maximal Codes Research Articles

Factorizing the polynomial of a code

We give an extension and a simplified presentation of a theorem of Schützenberger. This theorem describes the factorization of the commutative polynomial associated with a finite maximal code. It is the deepest result known so far in the theory of (variable-length) codes.

Finite State Models in the Study of Comma-Free Codes

The definition of comma-free codes is extended to be more comprehensive and include codes with variable length codewords and codes with an unlimited number of codewords. Finite state recognizers are used to represent commafree codes in a compact form, making available the well established methods of manipulating and analysing finite state models. An efficient algorithm is presented to test the comma-freeness of any given code represented by a finite state recognizer. It is well known, that any maximal comma-free code with codewords of odd fixed length can be achieved. Although upper limits have been given for even fixed length it has not been established which comma-free codes of this nature can achieve the upper limit. Compact representations of comma-free binary codes are given for variable length codewords with maximum even fixed lengths (6, 8 and 10) and in one case (length not greater than 6) it is seen that the corresponding upper limit for fixed lengths can be exceeded.

The enumeration of certain run length sequences

Binary block codes of length n with the property that an arbitrary catenation of codewords will produce a sequence with no runs of either zeros or ones of length greater than k , are considered. The enumeration of maximal codes is determined and the maximal rate such a code may have is found as the logarithm to the base two of the largest real root of a given polynomial.

On the triangle conjecture

We prove a noncommutative version of a theorem of Schützenberger on the factorization of variable-length codes. As consequences, we obtain a positive answer to a weak form of the `factorization conjecture, a complete characterization of maximal and finite codes and a noncommutative extension of an invariance property due to Hansel and Perrin.

Construction of error correcting codes by interpolation

A generalized and unified method of interpolation and transformation is used to generate all known maximal distance codes and important subfield subcodes. Some powerful tools for the analysis and synthesis of maximal distance codes are presented, as well as a generalization of the Mattson-Solomon polynomial and Lagrange and Fourier transforms to more general functions. In certain cases new codes can be obtained by differentiating a kernel function. Some further generalizations of Srivastava codes are constructed. A general method of decoding is given which can be used for complete decoding of ali coset leaders.

Maximal Prefix-Synchronized Codes

A prefix-synchronized code with prefix P of length p is a collection of strings whose first p characters equal P, but which have the property that P does not occur in any concatenation of codewords in any positions but those corresponding to initial segments of codewords. Given a block length N and the size q of the alphabet, it is desirable to find prefixes which maximize the size of the code. Gilbert conjectured that if $q = 2$, some maximal prefix-synchronized codes have prefixes of the form $11 \ldots 10$ for a suitable length p. It is shown here that this conjecture is true (at least for large N) for $q = 2$, as well as $q = 3$ and $q = 4$, but that it is false for $q \geqq 5$. It is also shown that the sizes of maximal prefix-synchronized codes exhibit some rather interesting oscillations as the block length increases. The proofs involve combinatorial arguments to establish closed form expressions for generating functions as well as extensive analysis of those generating functions.

The weight distribution of irreducible cyclic codes with block lengths [formula omitted

We study the weight distribution of irreducible cyclic ( n, k) codeswith block lengths n = n 1(( q 1 − 1)/ N), where N| q − 1, gcd( n 1, N) = 1, and gcd( l, N) = 1. We present the weight enumerator polynomial, A( z), when k = n 1 l, k = ( n 1 − 1) l, and k = 2 l. We also show how to find A( z) in general by studying the generator matrix of an ( n 1, m) linear code, V ∗ d over GF( q d ) where d = gcd ( ord n 1( q), l). Specifically we study A( z) when V ∗ d is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A μ which completely determine the weight distributionof any irreducible cyclic ( n 1(2 1 − 1), k) code over GF(2) for all n 1 ⩽ 17.

Self-Dual Codes over ${\text{GF}}( 3 )$

This paper studies self-dual and maximal self-orthogonal codes over ${\text{GF}}( 3 )$. First, a number of Gleason-type theorems are given, describing the weight enumerators of such codes. Second, a table of all such codes of length $\leqq 12$ is constructed. Finally, the complete weight enumerators of various quadratic residue and symmetry codes of length $\leqq 60$ are obtained.

The comma-free codes with words of length two

A code not requiring a distinct symbol to separate words is called comma-free. Two codes are isomorphic if one can be obtained from the other by a permutation of the underlying alphabet. Since subcodes of comma-free codes are comma-free, we investigate only maximal comma-free codes. All isomorphism classes of maximal comma-free codes with words of length 2 are determined and a natural representative of each class is given.

Maximal<tex>q</tex>-nary linear codes with large minimum distance (Corresp.)

This correspondence presents a complete solution to the problem of constructing maximal q -nary codes, for any word length n and minimum distance d in the region qd \geq (q - 1)n . Several interesting results regarding the interdependence of these codes are developed, which reveal the unique structure of the code spaces of the region qd \geq (q - 1)n . Some new maximal codes are obtained for the region qd .

On maximal comma-free codes (Corresp.)

In this correspondence a binary maximal comma-free code for words of length 10 and the search technique used to construct it are described.

A General Class of Maximal Codes ror Computer Applications

The error-correcting codes for symbols from GF (2b) are often used for correction of byte-errors in binary data. In these byte-error-correcting codes each check symbol in GF (2b) is expressed as b binary check digits and each information symbol in GF (2b), likewise, is expressed by b binary information digits. A new class of codes for single-byte-error correction is presented. The code is general in that the code structure does not depend on symbols from GF (2b). In particular, the number of check bits are not restricted to the multiples of b as in the case of the codes derived from GF (2b) codes. The new codes are either perfect or maximal and are easily implementable using shift registers.

Maximal Group Codes with Specified Minimum Distance

All n-digit maximal block codes with a specified minimum distance d such that 2d ≥ n can be constructed from the Hadamard matrices. These codes meet the Plotkin bound. In this paper we construct all maximal group codes in the region 2d ≥ n, where d is a specified minimum distance and n is the number of digits per code word. Unlike the case of block codes, the Plotkin upper limit, in general, fails to determine the number of code words B(n, d) in a maximal group code in the region 2d ≥ n. We show that the value of B(n, d) largely depends on the binary structure of the number d. An algorithm is developed that determines B(n, d), the maximum number of code words for given d and n ≤ 2d. The maximal code is, then, given by its modular representation, explicitly in terms of certain binary coefficients and constants related to n and d. As a side result, we obtain a new upper bound on the number of code words in the region 2d < n which is, in general, stronger than Plotkin's extended bound.

Analog Matched Filter Using Tapped Accoustic Surface Wave Delay Line (Correspondence)

VHF experiments are described employing 11- and 50-tap surface acoustic wave delay lines fabricated on crystalline quartz as analog matched filters. Excellent correlation characteristics were observed for a 63-bit maximal length biphase code.

The capacity of the product channel

Given any pair of arbitrary alphabet channels satisfying coding theorem and its strong converse. Then the coding theorem and its strong converse also holds for the product channel and the capacity of the product channel equals the sum of the capacities of the components. Wyner derives this result in (Wyner, 1966) by means of list decoding methods (which obviously have no strong connection with this kind of problem). Here a simpler proof of the statement above is presented which uses only ‘maximal code estimate’ (see Wolfowitz, 1964) and a converse estimate.

On the uniqueness of the Golay codes

A method is established to convert results from finite geometries to error-correcting codes. Using this we can determine the dimension of a maximal self-orthogonal code and the number of self-orthogonal codes of any fixed dimension in a given space. These results are applied to the Golay (11,6) code over GF(3) giving five equivalent conditions for a code to be equivalent to the Golay code. One of these is that every perfect (11,6) code is equivalent to the Golay code. Analogous conditions are established for the codes equivalent to the extended Golay (12,6) code over GF(3). One of these is that every (12,6) code with minimum weight 6 is equivalent to the Golay code. The corresponding theorems for the Golay (23,12) code over GF(2) and the extended Golay (24,12) code over GF(2) are given. Since these results are easier to demonstrate than the results over GF(3), the method of proof here uses known facts about these codes.

On the construction of comma-free codes

A construction is given which produces maximal (in number of words) comma-free codes for any odd word length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and any alphabet size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma</tex> .

Multiplexing using quasiorthogonal binary functions

The process of correlation detection is briefly reviewed and the properties of binary codes are stated for the purpose of utilizing these codes as a quasiorthogonal multiplexing function. The maximal linear code is shown to exhibit the best properties, particularly those generated by a Mersenne prime. The relationship between system bandwidth and code length is derived.

A note on a result in the theory of code construction

This note establishes a connection between Hadamard matrices H 4 t and the maximal binary codes M (4 t , 2 t ; 8 t ), M (4 t — 1, 2 t ; 4 t ) and M (4 t — 2, 2 t ; 2 t ) in two symbols 0 and 1, where by M ( n , d ; m ) we mean a set of m n -place sequences with 0 and 1 such that the Hamming distance between any two sequences is greater than or equal to d . The structure of these maximal codes is also studied in this paper.