On the structure of linear codes with covering radius two and three

We obtain new bounds on l(m,r), the minimum length of a linear code with codimension m and covering radius r. All bounds are derived in a uniform way. We employ results from coding theory, some earlier results on covering codes, and combinatorial arguments. We prove the following bounds: l(6, 2)=13, l(7,2)=19, l(8,2)⩾25, l(9,2)⩾34, l(2m-l,2)⩾2^m+1 for m⩾3, l(14,2)⩾182, l(16,2)⩾363, l(18,2)⩾725, l(20,2)⩾1449, l(22,2)⩾2897, l(24,2)⩾5794, l(9,3)⩾17, l(10,3)⩾21, l(12,3)⩾31, l(13,3)⩾38