encoding, power,

We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider F-linear properties that are invariant under linear transformations of the domain and prove that an O(1)-local “characterization ” is a necessary and sufficient condition for O(1)-local testability when |K | = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)-local constraint implies O(1)-local testability. These results generalize and extend the class of algebraic properties, most notably linearity and low-degree-ness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)-locally testable. Our results are proved by introducing a new notion that we term “formal characterizations”. Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations

We obtain improved bounds for the minimum weight of the dual codes associated with the codes from finite geometries in the case of odd order, and some results that apply also to the dual codes of non-desarguesian planes of odd order.

We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed-Muller and algebraic-geometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound.

An explicit basis of incidence vectors for the p-ary code of the design of points and hyperplanes of the affine geometry AGm (F p ) for any prime p and any integer m # 2 is obtained, which, as a corollary, gives a new elementary proof that this code is a generalized Reed-Muller code. In the proof a class of non-singular matrices related to Vandermonde matrices is introduced.

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraic-geometric. In particular, we investigate the family of q-ary Hamming codes, and prove that only those with redundancy one or two, and the binary [7; 4; 3] code are algebraic-geometric in this sense. For these codes we explicitly give a curve, rational points and a divisor. We prove that this triple is in a certain sense unique in the case of the [7; 4; 3] code. Key words: algebraic-geometric codes, algebraic curves, divisors, generalized Goppa codes, geometric Goppa codes. I. Introduction Since the early papers by Goppa [5],[6],[7], [8], algebraic-geometric codes have been in the spotlight of coding theoretic research for about a decade. As is well-known, numerous exciting results have been achieved using Goppa's construction of li...

Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1

We establish the range of values of #, where 0 1), for which the generalized Reed-Muller code RF q (#, m) of length q over the field F q of order q is spanned by its minimum-weight vectors.

We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fq-rational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Gröbner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.

Abstract—A maximum a posteriori (MAP) probability decoder of a block code minimizes the probability of error for each transmitted symbol separately. The standard way of implementing MAP decoding of a linear code is the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm, which is based on a trellis representation of the code. The complexity of the BCJR algorithm for the first-order Reed–Muller (RM-1) codes and Hamming codes is proportional to, where is the code’s length. In this correspondence, we present new MAP decoding algorithms for binary and nonbinary RM-1 and Hamming codes. The proposed algorithms have complexities proportional to �� � , where is the alphabet size. In particular, for the binary codes this yields complexity of order �� �. Index Terms—Hamming codes, maximum a posteriori (MAP) decoding, Reed–Muller codes. I.