An alternative to the Hamming code in the class of SEC-DED codes in semiconductor memory

A Family of Complete Caps in PG (n, 2)

“…Davydov and Tombak [5] also proved that, for any n ≥ 3 and any k subject to 5 · 2 n−3 ≥ k ≥ max{5 · 2 n−3 − 8, 9 · 2 n−4 − 1, 17 · 2 n−5 + 1}…”

“…Davydov and Tombak [5] proved that, for any k ≥ 0, the cap 2 k X (which is X k+3 in the notation of [1]) is optimal. The dual codes of the caps 2 k X are called " codes" in [5]. The codes were proposed by Panchenko in 1981, see the reference in [5].…”

“…Moreover, the caps in this sequence are the 'least dependent' among all caps of the same size in the same space, where the dependency of a cap is measured by the number of linearly dependent quadruples of points in it. (See [5] or the beginning of Section 3.1 below for a further discussion of this statement.) The main purpose of this article is to generalize the complete 5-cap in PG(3,2) (which becomes L 1 or L 4,0 in our notation) to a one-parameter family (L k ) and a two-paramter family (L r,s ) of complete caps.…”

“…Basically, the condition (3) allows one to remove up to 8 points fram 2 n−3 L 1 , while avoiding comparisons against doublings of other critical caps. Theorem 2 and Algorithm 1 in [5] describe the sequence of paints to be removed.…”

“…The dual codes of the caps 2 k X are called " codes" in [5]. The codes were proposed by Panchenko in 1981, see the reference in [5]. By the structure theorem of large complete caps, for proving the optimality of 2 k X = 2 k L 1 it is sufficient to show that for any subset C of the complete affine cap in PG(k + 3, 2) with size |C| = 5 · 2 k , A 4 (C) ≥ A 4 (2 k X) holds.…”