Best Known (31, 35, s)-Nets in Base 2
(31, 35, 131073)-Net over F2 — Constructive and digital
Digital (31, 35, 131073)-net over F2, using
(31, 35, 131090)-Net over F2 — Digital
Digital (31, 35, 131090)-net over F2, using
- net defined by OOA [i] based on linear OOA(235, 131090, F2, 4, 4) (dual of [(131090, 4), 524325, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(235, 131090, F2, 3, 4) (dual of [(131090, 3), 393235, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(235, 131090, F2, 4) (dual of [131090, 131055, 5]-code), using
- 1 times truncation [i] based on linear OA(236, 131091, F2, 5) (dual of [131091, 131055, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(235, 131072, F2, 5) (dual of [131072, 131037, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(218, 131072, F2, 3) (dual of [131072, 131054, 4]-code or 131072-cap in PG(17,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(236, 131091, F2, 5) (dual of [131091, 131055, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(235, 131090, F2, 4) (dual of [131090, 131055, 5]-code), using
- appending kth column [i] based on linear OOA(235, 131090, F2, 3, 4) (dual of [(131090, 3), 393235, 5]-NRT-code), using
(31, 35, 262141)-Net in Base 2 — Upper bound on s
There is no (31, 35, 262142)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 34359 869438 > 235 [i]