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In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [ABSTRACT FROM AUTHOR]

Let A (n , d) be the size of the maximum binary code of length n and minimum Hamming distance d. A (n , d , w) is defined similarly for binary code with constant weight w. Obviously, finding the value of A (n , d) is equivalent to finding the maximum independent set of the d − 1 th-power of n -dimensional hypercube. Based on this, this paper obtains that A (3 m , 2 m) = 4 , A (m (m + 1) 2 , 2 m − 2 , m) = m + 1 , A (2 m + 1 − 2 , 2 m − 1) = 2 m + 1 , and explores the structure of the maximum code of length 2 m + 2 and minimum Hamming distance 2 m + 1 , where m is an integer. [ABSTRACT FROM AUTHOR]

Self-dual and maximal self-orthogonal codes over GF (q) , where q is even or q ≡ 1 (mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator W C ¯ (x , y) of a self-dual code C ¯ is given in terms of a maximal self-orthogonal code C , where C ¯ is obtained by the extension of C. [ABSTRACT FROM AUTHOR]

In this paper, we introduce self-dual cyclic codes over the ring ℤ 2 × (ℤ 2 + u ℤ 2) = ℤ 2 ℤ 2 [ u ]. We determine the conditions for any ℤ 2 ℤ 2 [ u ] -cyclic code to be self-dual, that is, 𝒞 = 𝒞 ⊥ . Since the binary image of a self-orthogonal ℤ 2 ℤ 2 [ u ] -linear code is also a self-orthogonal binary linear code, we introduce quantum codes over ℤ 2 ℤ 2 [ u ]. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters. [ABSTRACT FROM AUTHOR]

Etzion [T. Etzion, Perfect byte correcting codes, IEEE Trans. Inform. Theory44 (1998) 3140–3146.] has classified byte error correcting binary codes into five different categories with respect to the size of the byte. If a code is partitioned into m equal byte of size say β then n = m β or the size of bytes is n m . Alternatively if bytes are of different size say, β 1 , β 2 , ... , β m then n = ∑ i = 1 m β i . The result was further modified by Tyagi and Sethi [V. Tyagi and A. Sethi, b i -Byte correcting perfect codes, Asian-Eur. J. Math.7(1) (2014) 1–8.] and the classification of bytes was given with respect to size of the byte as well as length of the burst. We call such codes as b i -byte correcting perfect codes. Our aim in this paper is to find the possibilities for the existence of b i -byte correcting non-binary perfect codes. [ABSTRACT FROM AUTHOR]

ℤ 2 ℤ 2 s -additive codes for any integer s > 1 are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over ℤ 2 s . In this paper, we are interested in studying ℤ 2 ℤ 2 s -additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable ℤ 2 ℤ 2 s -additive codes and provide conditions on the generator polynomials for a ℤ 2 ℤ 2 s -additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of ℤ 2 ℤ 8 -additive cyclic codes. [ABSTRACT FROM AUTHOR]

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485-505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph and find PD-sets for all values of and . Further we show that these PD-sets are minimal when is odd and . [ABSTRACT FROM AUTHOR]

For every Hadamard design with parameters 2 - (n − 1 , n 2 − 1 , n 4 − 1) having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters 2 - (4 n − 1 , 2 n − 1 , n − 1). Moreover, for the case n = 8 we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II ℤ 4 -codes of length 32. The constructed codes are the first examples of extremal Type II ℤ 4 -codes of length 32 and type 4 k 1 2 k 2 , k 1 ∈ { 7 , 8 , 9 , 1 0 } , whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II ℤ 4 -codes of length 32 and type 4 1 4 2 4 . [ABSTRACT FROM AUTHOR]