MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.Comment: 12 page

The extension problem for Lee and Euclidean weights

The extension problem is solved for the Lee and Euclidean weights over three families of rings of the form $\Z/N\Z$: $N=2^{\ell + 1}$, $N=3^{\ell + 1}$, or $N=p=2q+1$ with $p$ and $q$ prime. The extension problem is solved for the Euclidean PSK weight over $\Z/N\Z$ for all $N$

Self-Dual Codes over Commutative Frobenius Rings

We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for finite commutative Frobenius rings, showing that all self-dual codes with minimum weight greater than 2 can be obtained in this manner in cases where the construction applies. Key Words: Self-dual codes, codes over rings. 2

Generating and Searching Families of FFT Algorithms

A fundamental question of longstanding theoretical interest is to prove the lowest exact count of real additions and multiplications required to compute a power-of-two discrete Fourier transform (DFT). For 35 years the split-radix algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic operations on real numbers for a size-n DFT, and was widely believed to be the best possible. Recent work by Van Buskirk et al. demonstrated improvements to the split-radix operation count by using multiplier coefficients or "twiddle factors" that are not n-th roots of unity for a size-n DFT. This paper presents a Boolean Satisfiability-based proof of the lowest operation count for certain classes of DFT algorithms. First, we present a novel way to choose new yet valid twiddle factors for the nodes in flowgraphs generated by common power-of-two fast Fourier transform algorithms, FFTs. With this new technique, we can generate a large family of FFTs realizable by a fixed flowgraph. This solution space of FFTs is cast as a Boolean Satisfiability problem, and a modern Satisfiability Modulo Theory solver is applied to search for FFTs requiring the fewest arithmetic operations. Surprisingly, we find that there are FFTs requiring fewer operations than the split-radix even when all twiddle factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on Satisfiability, Boolean Modeling and Computatio