Best Known (78, 93, s)-Nets in Base 2
(78, 93, 1172)-Net over F2 — Constructive and digital
Digital (78, 93, 1172)-net over F2, using
- net defined by OOA [i] based on linear OOA(293, 1172, F2, 15, 15) (dual of [(1172, 15), 17487, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(293, 8205, F2, 15) (dual of [8205, 8112, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 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(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(293, 8205, F2, 15) (dual of [8205, 8112, 16]-code), using
(78, 93, 2051)-Net over F2 — Digital
Digital (78, 93, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(293, 2051, F2, 4, 15) (dual of [(2051, 4), 8111, 16]-NRT-code), using
- OOA 4-folding [i] based on linear OA(293, 8204, F2, 15) (dual of [8204, 8111, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 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(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- OOA 4-folding [i] based on linear OA(293, 8204, F2, 15) (dual of [8204, 8111, 16]-code), using
(78, 93, 30561)-Net in Base 2 — Upper bound on s
There is no (78, 93, 30562)-net in base 2, because
- 1 times m-reduction [i] would yield (78, 92, 30562)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4952 634479 829232 759353 195152 > 292 [i]