Limited magnitude error model has applications in flash memory. In this model, a perfect code is equivalent to a tiling of$\mathbb {Z}^{n}$by limited magnitude error balls. In this paper, we give a complete classification of lattice tilings of$\mathbb {Z}^{n}$by limited magnitude error balls$\mathcal {B}(n,2,1,1)$.

For an $r$-graph $G$, the minimum $(r-1)$-degree $\delta(G)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(G)$ is contained in at least $t$ edges of $G$. Given an $r$-graph $F$, the codegree density $\gamma(F)$ is the largest $\gamma>0$ such that there are $F$-free $r$-graphs $G$ on $n$ vertices with $\delta(G)\ge(\gamma-o(1))n$. In this paper, we consider the codegree density of projective geometries. Employing the moment identity of a subset of $PG_{m}(q)$, we prove (1) $\gamma(PG_{2}(q))=\frac{1}{2}$ for prime power $q\equiv2\pmod{3}$; and (2) $\gamma(PG_{3}(q))=\frac{2}{3}$ for prime power $q\equiv1\pmod{2}$ or $q\equiv2\pmod{3}$. Our results partially solve an open problem proposed by Keevash and Zhao [ J. Combin. Theory Ser. B, 97 (2007), pp. 919--928]. Previously, the codegree density problems for projective geometries were settled only for $PG_{2}(2)$, $PG_{3}(2)$, $PG_{3}(3),$ and $PG_{2}(q)$ with odd prime power $q$.

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn [ Repeated Patterns in Proper Colourings, preprint, https://arxiv.org/abs/2002.00921 (2020)]. Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a proper edge coloring of the complete graph $K_{n}$ with $c$ colors containing no $k$ vertex-disjoint color-isomorphic copies of $H$. Using algebraic properties of polynomials over finite fields, we give an explicit proper edge coloring of $K_{n}$ and show that $f_{k}(n, C_{4})=\Theta(n)$ when $k\geqslant 3$ and $n\rightarrow\infty$. The methods we used in the edge coloring may be of some independent interest. We also consider a related generalized Ramsey problem. For given graphs $G$ and $H,$ let $r(G,H,q)$ be the minimum number of edge colors (not necessarily proper) of $G$, such that the edges of every copy of $H\subseteq G$ together receive at least $q$ distinct colors. Establishing the relation to the Turán number of specified bipartite graphs, we obtain some general lower bounds for $r(K_{n,n},K_{s,t},q)$ with a broad range of $q$.

Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of nonsingular perfect splitter sets, B[-k1, k2](p) sets, where 0 ≤ k1≤ k2= 4. Meanwhile, constructions of nonsingular perfect splitter sets are given. When perfect splitter sets do not exist, we present four new constructions of quasi-perfect splitter sets. Finally, we give a connection between nonsingular splitter sets and Cayley graphs, and as a byproduct, a general lower bound on the maximum size of nonsingular splitter sets is given.

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict avoiding codes. In this paper, we prove some nonexistence results for nonsingular perfect splitter sets. We also give some necessary conditions for the existence of purely singular perfect splitter sets. Finally, we apply these results to purely singular perfect B[-1, k](m) and B[-2, k](m) sets for small k. In particular, we solve completely the problems left by Schwartz (European J. Combin., vol. 36, pp.130-142, Feb. 2014).

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect r error correcting Lee code of length n for n ≥ 3 and r > 1. In this paper, we study perfect codes in Zn under the l p metric, where 11/p, 31/p. We also give an algebraic construction of quasi-perfect lpcodes for p = 1, r = 2, and 2 <; p <; ∞, r = 21/p.

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict-avoiding codes. The study of k-radius sequences was motivated by some problems occurring in large data transfer. It is observed that the existence of splitter sets yields k-radius sequences of short length. In this paper, we obtain several new results contributing to splitter sets and k-radius sequences. We give some new constructions of perfect splitter sets, as well as some nonexistence results on them. As a byproduct, we obtain some new results on optimal conflict-avoiding codes. Furthermore, we provide several explicit constructions of short k-radius sequences for certain values of n, by establishing the existence of k-additive sequences. In particular, we show that for any fixed k, there exist infinitely many values of n such that fk(n) = 2k/n2+ O(n), where fk(n) denotes the shortest length of an n-ary k-radius sequence. This result partially affirms a conjecture posed by Bondy, Lonc, and Rza̧żewski.

Some physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This motivated the application of the asymmetric limited magnitude error model in flash memory. In this paper, we present a new construction of quasi-perfect codes for such errors, and we also study the perfect codes with symmetric errors. Moreover, we show some nonexistence results on perfect codes for correcting single error of limited magnitude.

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature.

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes.

Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. In this paper, using Hermitian construction and classical constacyclic codes, we construct six classes of quantum MDS codes. Two of these six classes of quantum MDS codes have larger minimum distance than the ones available in the literature. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

The determination of the cross correlation between an m-sequence and its decimated sequence has been a longstanding research problem. Considering a ternary m-sequence of period 33r- 1, we determine the cross correlation distribution for decimations d = 3r+ 2 and d = 32r+ 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 22lm- 1, we make an initial investigation for the decimation d = (22lm- 1)/(2m+ 1) + 2s, where l ≥ 2 is even and 0 <; s <; 2m - 1. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.