New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R = 2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r = 2k + 1 (the case q = 3, r = 4k + 1 was considered earlier). New code families with r = 4k are also obtained. An updated table of upper bounds on the length function for linear codes with r ≤ 24, R = 2, and q = 3, 5 is given.