Best Known (78−13, 78, s)-Nets in Base 8
(78−13, 78, 349526)-Net over F8 — Constructive and digital
Digital (65, 78, 349526)-net over F8, using
- net defined by OOA [i] based on linear OOA(878, 349526, F8, 13, 13) (dual of [(349526, 13), 4543760, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(878, 2097157, F8, 13) (dual of [2097157, 2097079, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 2097159, F8, 13) (dual of [2097159, 2097081, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(878, 2097159, F8, 13) (dual of [2097159, 2097081, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(878, 2097157, F8, 13) (dual of [2097157, 2097079, 14]-code), using
(78−13, 78, 1470768)-Net over F8 — Digital
Digital (65, 78, 1470768)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(878, 1470768, F8, 13) (dual of [1470768, 1470690, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using
(78−13, 78, large)-Net in Base 8 — Upper bound on s
There is no (65, 78, large)-net in base 8, because
- 11 times m-reduction [i] would yield (65, 67, large)-net in base 8, but