Best Known (97, 97+12, s)-Nets in Base 2
(97, 97+12, 43693)-Net over F2 — Constructive and digital
Digital (97, 109, 43693)-net over F2, using
- net defined by OOA [i] based on linear OOA(2109, 43693, F2, 12, 12) (dual of [(43693, 12), 524207, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2109, 262158, F2, 12) (dual of [262158, 262049, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 262163, F2, 12) (dual of [262163, 262054, 13]-code), using
- 1 times truncation [i] based on linear OA(2110, 262164, F2, 13) (dual of [262164, 262054, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(291, 262144, F2, 11) (dual of [262144, 262053, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(2110, 262164, F2, 13) (dual of [262164, 262054, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 262163, F2, 12) (dual of [262163, 262054, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2109, 262158, F2, 12) (dual of [262158, 262049, 13]-code), using
(97, 97+12, 65540)-Net over F2 — Digital
Digital (97, 109, 65540)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2109, 65540, F2, 4, 12) (dual of [(65540, 4), 262051, 13]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2109, 262160, F2, 12) (dual of [262160, 262051, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 262163, F2, 12) (dual of [262163, 262054, 13]-code), using
- 1 times truncation [i] based on linear OA(2110, 262164, F2, 13) (dual of [262164, 262054, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(291, 262144, F2, 11) (dual of [262144, 262053, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(2110, 262164, F2, 13) (dual of [262164, 262054, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 262163, F2, 12) (dual of [262163, 262054, 13]-code), using
- OOA 4-folding [i] based on linear OA(2109, 262160, F2, 12) (dual of [262160, 262051, 13]-code), using
(97, 97+12, 880905)-Net in Base 2 — Upper bound on s
There is no (97, 109, 880906)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 649 037895 972870 169814 875210 072196 > 2109 [i]