Best Known (78, 78+15, s)-Nets in Base 8
(78, 78+15, 299595)-Net over F8 — Constructive and digital
Digital (78, 93, 299595)-net over F8, using
- net defined by OOA [i] based on linear OOA(893, 299595, F8, 15, 15) (dual of [(299595, 15), 4493832, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(893, 2097166, F8, 15) (dual of [2097166, 2097073, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(893, 2097167, F8, 15) (dual of [2097167, 2097074, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(81, 15, F8, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(893, 2097167, F8, 15) (dual of [2097167, 2097074, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(893, 2097166, F8, 15) (dual of [2097166, 2097073, 16]-code), using
(78, 78+15, 1992530)-Net over F8 — Digital
Digital (78, 93, 1992530)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(893, 1992530, F8, 15) (dual of [1992530, 1992437, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(893, 2097167, F8, 15) (dual of [2097167, 2097074, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(81, 15, F8, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(893, 2097167, F8, 15) (dual of [2097167, 2097074, 16]-code), using
(78, 78+15, large)-Net in Base 8 — Upper bound on s
There is no (78, 93, large)-net in base 8, because
- 13 times m-reduction [i] would yield (78, 80, large)-net in base 8, but