Best Known (113−14, 113, s)-Nets in Base 2
(113−14, 113, 9364)-Net over F2 — Constructive and digital
Digital (99, 113, 9364)-net over F2, using
- net defined by OOA [i] based on linear OOA(2113, 9364, F2, 14, 14) (dual of [(9364, 14), 130983, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2113, 65548, F2, 14) (dual of [65548, 65435, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 65553, F2, 14) (dual of [65553, 65440, 15]-code), using
- 1 times truncation [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 65553, F2, 14) (dual of [65553, 65440, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2113, 65548, F2, 14) (dual of [65548, 65435, 15]-code), using
(113−14, 113, 16388)-Net over F2 — Digital
Digital (99, 113, 16388)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2113, 16388, F2, 4, 14) (dual of [(16388, 4), 65439, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2113, 65552, F2, 14) (dual of [65552, 65439, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 65553, F2, 14) (dual of [65553, 65440, 15]-code), using
- 1 times truncation [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2113, 65553, F2, 14) (dual of [65553, 65440, 15]-code), using
- OOA 4-folding [i] based on linear OA(2113, 65552, F2, 14) (dual of [65552, 65439, 15]-code), using
(113−14, 113, 244559)-Net in Base 2 — Upper bound on s
There is no (99, 113, 244560)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10384 647186 870854 748635 055744 973453 > 2113 [i]