p-Adic estimates of hamming weights in abelian codes over galois rings

Abstract

Abstract

Point count divisibility for algebraic sets over ℤ/𝕡^{ℓ}ℤ and other finite principal rings

“…, f t to be homogeneous, but the proof that he supplies only establishes the above proposition as stated, for he reduces the general case to the case t = 1 and then relies on Ax's proof [3] of the t = 1 case, which unfortunately does not always provide a homogeneous polynomial. However, it is not difficult to modify Ax's proof of the t = 1 case to invariably furnish homogeneous polynomials and thus to substantiate N. M. Katz's claim (see [9] for such a modification). Although Proposition 1.3 shows that the lower bounds on p-divisibility established by Ax and Katz cannot be improved if we are given only the degrees of the polynomials, there are many improvements and extensions that concern polynomials of specific forms or use information other than or in addition to the degrees of the polynomials [20], [8], [1], [24], [13], [7], [18], [14], [22], [21], [16], [15], [19], [17], [4], [5].…”