Error-Correction of Multidimensional Bursts

A construction for D-dimensional binary codes of size n
₁
timesn
₂
timeshelliptimesn
_D
correcting a single D-dimensional box error is presented. If the size of the box error is b
₁
timesb
₂
timeshelliptimesb
_D
, b
_i
odd, 1les i les D, and B = Pi
_i

^D
=
₁
b
_i
, then the redundancy of the code is at most [log
₂
(n
₁
n
₂
hellip n
_D
)] +B + (D-2)[log
₂
B] + [log
₂
b
₁
]. For a two-dimensional binary array of size n times n we present a code correcting an error whose shape is a Lee sphere with radius R. The redundancy of the code is at most [log
₂
n
²
] + 2R
²
+ 2R + [2log
₂
(2R+1)]+1. This is also the redundancy of a binary code which corrects an arbitrary two-dimensional cluster-error of size 2R+1. A generalization for D-dimensional code which corrects either D-dimensional error whose shape is a Lee sphere or an arbitrary cluster-error is also given.