The notion of algebraic immunity of Boolean functions has been generalized in several ways to vector-valued functions and/or over arbitrary finite fields and reasonable upper bounds for such generalized algebraic immunities has been proved in Armknecht and paper we show that the upper bounds can be reached as the maximal values of algebraic immunities for most of generalizations by using properties of Reed-Muller codes.
The complexity of decoding the standard Reed-Solomon code is a well-known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, Li and Wan showed that the error distance can be determined when the degree of the received word as a polynomial is small. In the first part, the result of Li and Wan is improved. On the other hand, one of the important parameters of an error-correcting code is the dimension. In most cases, one can only get bounds for the dimension. In the second part, a formula for the dimension of the generalized trace Reed-Solomon codes in some cases is obtained.
In this paper, deep holes of Reed-Solomon (RS) codes are studied. A new class of deep holes for generalized affine RS codes is given if the evaluation set satisfies certain combinatorial structure. Three classes of deep holes for projective Reed-Solomon (PRS) codes are constructed explicitly. In particular, deep holes of PRS codes with redundancy three are completely obtained when the characteristic of the finite field is odd. Most (asymptotically of ratio 1) of the deep holes of PRS codes with redundancy four are also obtained.
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