We study the binary linear code families associated with certain types of multivariable functions. We observe that a majority of these codes are not optimal codes nor few weight codes yet. In this paper, we find infinite families offew weight(near-)optimalbinary linear codes from our code families. Furthermore, we produce supportt-designs (t= 2 or 3) which cannot be determined by theAssmus-Mattson Theorem; this is the first time that the result by Tang et al. was successfully used to prove that linear codes holdt-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using theCSS construction. As a main method, we use themodified shortening method(simply, calledshortening method), which is applied to our code families. Using the results on the weight distributions of our shortened codes, we verify that our codes families supportt-designs (t= 2, 3).We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.

It is important to study constructions of plateaued functions with balancedness and high algebraic degrees for preventing cryptographic attacks. Our goal of this paper is to find practical construction methods for producing infinite families of balanced$r$-plateaued functions with maximal algebraic degrees for every positive integer$r$. We first present a theoretical framework for secondary constructions of plateaued functions. From this framework, we derive three practical algorithms by controlling initial input vectors. These algorithms produce$(s+1)$-plateaued functions from a given bent function and$s$-plateaued functions in a recursive way for any nonnegative integer$s$; therefore, we obtain$r$-plateaued functions for every$r > s$. Then we obtain three concrete construction methods of balanced$r$-plateaued functions with maximal algebraic degrees from the algorithms. For implementation, in the tables, we list up some initial bent (0-plateaued) functions, which guarantee the maximality of algebraic degrees of plateaued functions. Furthermore, we discuss the complexities of the three algorithms, which shows the feasibility of our methods. We emphasize that this is the first time to give constructions of balanced$r$-plateaued functions with maximal algebraic degrees for every positive integer$r$as far as we know.

We construct several families of distance-optimal few-weight binary linear codes. As a method, we use the mixed alphabet ring Z2Z2[u],u2= 0 (viewing Z2Z2[u] as a Z2[u]-module) and three suitable defining sets, each consisting of three simplicial complexes generated by a single maximal element to construct three different families of linear codes over Z2[u],u2= 0. We explicitly determine their Lee weight distributions and study their Gray images to obtain our results. It turns out that most of the distance-optimal codes obtained in this paper are self-orthogonal and minimal as well. We emphasize that we find an infinite family of binary three-weight projective codes with new parameters, which produce stronglyl-walk-regular graphs for every oddl≥ 3.

Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed-Solomon type. Our method is based on the constructions of Beelen et al. (2017) and Roth and Lempel (1989). To the best of our knowledge, this is the first paper on the construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of non-Reed-Solomon type constructed by our method is not monomially equivalent to any LCD code constructed by the method of Carlet et al. (2018).

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔcconstructed from simplicial complexes in F2n, where Δ is a simplicial complex in F2nand Δcthe complement of Δ. We first find an explicit computable criterion for CΔcto be optimal; this criterion is given in terms of the 2-adic valuation of Σsj=12|Ai|-1, where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔcis a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of CΔc.We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

A p-ary function f in n variables is an l-form if f(tu) = tlf (u) for any nonzero t in Zpand u in Zpn. Let n be a positive even integer, p an odd prime, and l an element of {1, 2, . . . , p -1} provided that l ≠ p -1 if p > 3. Let f be a p-ary bent function in n variables of l-form with f (0) = 0 and gcd(l - 1, p - 1) = 1, and let Hl= {tl: t ∈ Zp*}. We denote by Gf,lthe Cayley graph Cay(Zpn, ∪s∈Hlf-1(s)). Our main results are as follows: 1) if there is weakly regular p-ary bent f which is not regular, then l is 2; 2) if l = 2, then f is weakly regular p-ary bent if and only if the Cayley graph G f,l is strongly regular; 3) if l ≠ 2, then f is regular p-ary bent if and only if the Cayley graph Gf,lis strongly regular; 4) Gf,lcan be replaced by Cay(Zpn, f-1(0)\{0}) in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of Gf,l.

We present three explicit methods for construction of formally self-dual codes over ℤ4. We characterize relations between Lee weight enumerators of formally self-dual codes of length n over ℤ4and those of length n t 2; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over ℤ4. Using these three constructions, we can find free formally selfdual codes over ℤ4, as well as non-free formally self-dual codes over ℤ4of all even lengths. We find free or non-free formally selfdual codes over ℤ4of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over ℤ4by using the Gray map.

Plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics. We find explicit criteria for the construction of p-ary r-plateaued functions with an odd prime p. We point out that 0-plateaued functions are bent functions, and so plateaued functions generalize the notion of bent functions. We first derive an explicit form for the Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upper bound on the degree of p-ary r-plateaued functions, and we classify p-ary (n - 1)-plateaued functions in n variables. We also obtain explicit criteria for the existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.

We find some necessary conditions for the existence of regular p-ary bent functions (from Znp to Zp), where p is a prime. In more detail, we show that there is no regular p-ary bent function f in n variables with w(Mf) larger than n/2, and for a given nonnegative integer k, there is no regular p-ary bent function f in n variables with w(Mf)=n/2-k ( n+3/2-k, respectively) for an even n ≥ Np,k(an odd n ≥ Np,k, respectively), where Np,kis some positive integer, which is explicitly determined and the w(Mf) of a p-ary function f is some value related to the power of each monomial of f. For the proof of our main results, we use some properties of regular p-ary bent functions, such as the MacWilliams duality, which is proved to hold for regular p-ary bent functions in this paper.

We classify extremal Hermitian self-dual quaternary codes of lengths 30 and 32 with an automorphism of odd prime order. We prove that there exists exactly one extremal Hermitian self-dual [30,15,12] quaternary code with a nontrivial automorphism of odd prime order, up to equivalence; the order of its automorphism group is 36 540. In fact, this code is equivalent to the extended quadratic residue code. We also prove that there exists no extremal Hermitian self-dual [32,16,12] quaternary code with a nontrivial automorphism of odd prime order.

We prove that MDS linear poset-codes satisfy Gilbert-Varshamov bound for their Hamming weights asymptotically. We also construct MDS linear poset-codes on arbitrary poset-metric spaces by using the Dilworth's chain decomposition theorem and results about the Hermite interpolation problem over a finite field. We prove that there exist linear poset-codes with large weights for both poset-metrics and Hamming metrics, as well.

The building-up construction for self-dual codes was developed by the authors over finite fields GF(q) when q is a power of 2 or q ¿ 1 (mod 4). In this paper, we complete the building-up construction for self-dual codes over GF(q) with q ¿ 3 (mod 4). For example, we construct new 945 extremal self-dual ternary [32, 16, 9] codes, each of which has a trivial automorphism group.

We construct new MDS or near-MDS self-dual codes over large finite fields. In particular, we show that there exists a Euclidean self-dual MDS code of length n = q over GF(q) whenever q = 2m(m ges 2) using a Reed-Solomon (RS) code and its extension. It turns out that this multiple description source (MDS) self-dual code is an extended duadic code. We construct Euclidean self-dual near-MDS codes of length n = q-1 over GF(q) from RS codes when q = 1 (mod 4) and q les 113. We also construct many new MDS self-dual codes over GF(p) of length 16 for primes 29 les p les 113. Finally, we construct Euclidean/Hermitian self-dual MDS codes of lengths up to 14 over GF(q2) where q = 19, 23,25, 27, 29.

In this paper we develop a complete generalization of the building-up method [J.-L. Kim, (2001)] for the Euclidean and Hermitian self-dual codes over finite fields GF(q). Using this method we construct many new Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF(q), where q=8, 9, 16, 25, 32, 41, 49, 53, 64, 81, and 128.