We consider lattice tilings of ℝ n by a shape we call a (k+, k-, n)-quasi-cross. Such lattices
form perfect error-correcting codes which correct a single limited-magnitude error with …

Error-correcting codes were introduced to combat errors in communication channels,
storage devices, and other computerized systems. In each such system, the information may …

A well-known conjecture of Golomb and Welch is that the only nontrivial perfect codes in the
Lee and Manhattan metrics have length two or minimum distance three. This problem and …

Since 1968, when the Golomb-Welch conjecture was raised, it has become the main motive
power behind the progress in the area of the perfect Lee codes. Although, there is a vast …

Lee codes have been intensively studied for more than 40 years. Interest in these codes has
been triggered by the Golomb-Welch conjecture on the existence of the perfect error …

Abstract The Golomb–Welch conjecture states that there are no perfect e-error-correcting
codes in Z n for n≥ 3 and e≥ 2. In this note, we prove the nonexistence of perfect 2-error …

In this paper we survey recent results on the Golomb–Welch conjecture and its
generalizations and variations. We also show that there are no perfect 2-error correcting Lee …

Abstract The Golomb–Welch conjecture deals with the existence of perfect e-error correcting
Lee codes of word length n, PL (n, e) codes. Although there are many papers on the topic …

Abstract In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with
radius r≥ 2 and dimension n≥ 3. A diameter perfect code is a natural generalization of the …

We study the problem of existence of (nontrivial) perfect codes in the discrete n n-simplex Δ
_ ℓ^ n:=\left {\left (x_0, ..., x_n): x_i ∈ Z _+, ∑ _i x_i= ℓ\right\} Δ ℓ n:= x 0,…, xn: xi∈ Z+,∑ ixi …