Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories

Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation

Near MDS poset codes and distributions

We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.Comment: 13 pages, 1 figur

A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added