A design-theoretic approach to binary linear codes that are single-error correcting (SEC) and double-error-detecting (DED) with the capability of detecting any error within a single byte of even width w (BED) is developed. A construction of SEC-BED-DED codes from ordinary binary linear codes utilizing partial Steiner systems is given. The construction produces some codes with higher rates than those known previously. The codes constructed may inherit special properties from the Steiner systems from which are derived. In particular, some rotational odd-weight-column codes are obtained.>

LetC(B)denote the binary cyclicANcode with generatorA, whereAB=2^{n} - 1. It is known thatC(B)is equidistant ifBis a prime powerp^{k}, where either2or-2is primitive moduloBprovidedp\equiv 1 \pmod{3}{\rm if} k > 1. It is conjectured that these are the onlyBsuch thatC(B)is equidistant. We have verified this forB < 100 000. Several results are established that further limit the possibilities for counterexamples to the conjecture.

In this correspondence we give a new definition of modular arithmetic weight (relative to any modulusm > 0and any radixr \geq 2), which has two very useful properties, i) It is defined for all integers and is invariant under congruence modulom. ii) It yields a metric for all modulim. In the important casesm = r^n, m = r^n - 1, andm =r^n + 1, it coincides with the modular weight of Rao and Garcia, and thus acts as a proper measure of errors for these moduli. Due to these properties, we are able to obtain conceptually simpler proofs of several known (as well as some new) results on the computation of modular distance in cyclic AN-codes. Along these same lines we introduce the notion of modular-cyclic nonadjacent form (NAF) whenm = r^n - 1. For moduli of this type, we show that every integer has a modular-cyclic NAF; the number of nonzero digits is the modular weight. It is a curious fact that a "modular" version of the well-known Chang/Tsao-Wu algorithm (for computing the NAF) turns out to yield the modular-cyclic NAF directly.

In this correspondence we define a "nonadjacent form" for integers in an arbitrary radixr > 1. This form is proved to be unique, and the arithmetic weight of an integer is shown to be equal to the number of nonzero terms in the form. Two algorithms are presented for the computation of this form. Ifr = 2, our form coincides with the well-known modified binary nonadjacent form.