We construct new self-dual and isodual codes over the integers modulo 4. The binary images of these codes under the Gray map are nonlinear, but formally self-dual. The construction involves Hensel lifting of binary cyclic codes. Quaternary quadratic residue codes are obtained by Hensel lifting of the classical binary quadratic residue codes. Repeated Hensel lifting produces a universal code defined over the 2-adic integers. We investigate the connections between this universal code and the codes defined over Z 4 , the composition of the automorphism group, and the structure of idempotents over Z 4 . We also derive a square root bound on the minimum Lee weight, and explore the connections with the finite Fourier transform. Certain self-dual codes over Z 4 are shown to determine even unimodular lattices, including the extended quadratic residue code of length q + 1, where q j \Gamma1(mod 8) is a prime power. When q = 23, the quaternary Golay code determines the Leech lattice in this way....

Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constant-weight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n 6 28 and d 6 14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n; d; w) by means of mapping constantweight codes into Euclidean space. This approach produces, among other results, a bound on A(n; d; w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-boundedweight codes, which may be thought of as a generaliz...

In this thesis we take a detailed look at the algebraic structure of convolutional and quasi-cyclic codes using the tools and methods of linear systems theory. Let F q be a finite field with q elements. In particular, we define convolutional codes as linear, right shift invariant, compact support behaviors in (F n ) Z+ . We then examine the concepts of observability, controllability, and minimality for convolutional codes as defined above. We show how convolutional codes are dual to the class of autoregressive behaviors. We compare compact support convolutional codes to non-compact support convolutional codes. In addition, we derive first order representations of convolutional codes on a purely module theoretic. We also examine the properties of these representations and give conditions for observability and minimality. Using the systems theoretic structure of convolutional codes we present two code constructions. For the first one we choose n; k; q and ffi 2 Z+ , such that q ffi ...

Let ( ) denote the maximum possible number of codewords in a binary code of length and minimum Hamming distance . For large values of , the best known upper bound, for fixed , is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of and , and for each there are infinitely many values of for which the new bound is better than the Johnson bound. For small values of and , the best known method to obtain upper bounds on ( ) is linear programming. We give new inequalities for the linear programming and show that with these new inequalities some of the known bounds on ( ) for 28 are improved.

A group is known as large if one of its finite index subgroups has a free nonabelian quotient. Large groups have many interesting properties, for example, super-exponential subgroup growth and infinite virtual first Betti number. It is therefore useful to be able to detect them in practice. In this paper, we will show

We study the task of transforming a hard function f, with which any small circuit disagrees on (1 − δ)/2 fraction of the input, into a harder function f ′ , with which any small circuit disagrees on (1 − δ k)/2 fraction of the input, for δ ∈ (0, 1) and k ∈ N. We show that this process can not be carried out in a black-box way by a circuit of depth d and size 2 o(k1/d) or by a nondeterministic circuit of size o(k / log k) (and arbitrary depth). In particular, for k = 2 Ω(n) , such hardness amplification can not be done in ATIME(O(1), 2 o(n)). Therefore, hardness amplification in general requires a high complexity. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently non-uniform in the following sense. Given as an oracle any algorithm which agrees with f ′ on (1 − δ k)/2 fraction of the input, we still need an additional advice of length Ω(k log(1/δ)) in order to compute f correctly on (1−δ)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f ′ , one has to start with a function f which is hard against non-uniform circuits. Finally, we derive similar lower bounds for any black-box construction of pseudorandom generators from hard functions.

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, including in optical code-division multiple-access communication systems and in constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback. In this paper we construct optimal constant weight cyclically permutable codes with length n, weight w, and a minimum Hamming distance 2w - 2. Some of these codes coincide with the well-known design called a difference family. Some of the constructions use combinatorial structures with other applications in coding.

Two classes of generalized concatenated (GC) codes with convolutional outer codes are studied. The first class is based on the classical Plotkin ja \Phi bjbj construction. A new suboptimal multi-stage soft decision algorithm is proposed and the corresponding performance bounds are obtained. These codes are shown to achieve better performance than conventional convolutional codes with equal or less decoding complexity, and are capable of unequal error protection. The Plotkin construction is then generalized using an inner differential encoding structure to obtain a second class of GC codes. A low-complexity two-iteration decoding algorithm using traditional hard-output Viterbi decoders is proposed. Numerical results show that the new coding systems can achieve comparable and sometimes superior performance to low-complexity turbo codes with similar computational complexity. Index Terms: Iterative decoding, concatenated codes, turbo codes. Revised draft of June 1, 1997 for IEEE Journal on...

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson-Hadamard matrices and codes over R meeting the Plotkin bound. Key Words: Frobenius ring, ring-linear codes, Plotkin bound, Butson-Hadamard matrix.

We prove that there are four designs F i such that each parallel class of any resolvable 2-(14; 7; 12) design having at least two blocks whose intersection has six points can be constructed as a pairwise union of two parallel classes of F l and (F r ), for some 2 S 14 . Then, with a parallel class by parallel class backtracking search we establish that there are exactly 541,192 nonisomorphic such 2-(14; 7; 12) designs. We also show that there are at least the same number of 3-(14; 7; 5) designs. Keywords: resolvable t-design; parallel class; backtracking search 1