On Multifold Packings of Radius-1 Balls in Hamming Graphs

A
$\lambda $
-fold
${r}$
-packing (multiple radius-
${r}$
covering) in a Hamming metric space is a code
${C}$
such that the radius-
${r}$
balls centered in the codewords of
${C}$
cover each vertex of the space by not more (not less, respectively) than
$\lambda $
times. The well-known
${r}$
-error-correcting codes correspond to the case
$\lambda =1$
, while in general multifold
${r}$
-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a
${q}$
-ary 2-fold 1-packing as
${q}$
grows; (b) prove that a
${q}$
-ary distance-2 MDS code of length
${n}$
is an optimal
${n}$
-fold 1-packing if
${q}\ge 2{n}$
; (c) derive an upper bound for the size of a binary
$\lambda $
-fold 1-packing and a lower bound for the size of a binary multiple radius-1 covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary 2-fold 1-packings up to length 9, in particular, establish the maximum size 96 of a binary 2-fold 1-packing of length 9; (e) prove some properties of 1-perfect unitrades, which are a special case of 2-fold 1-packings.