Best Known (78, 78+22, s)-Nets in Base 2
(78, 78+22, 152)-Net over F2 — Constructive and digital
Digital (78, 100, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 25, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
(78, 78+22, 257)-Net over F2 — Digital
Digital (78, 100, 257)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2100, 257, F2, 2, 22) (dual of [(257, 2), 414, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2100, 260, F2, 2, 22) (dual of [(260, 2), 420, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2100, 520, F2, 22) (dual of [520, 420, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2100, 521, F2, 22) (dual of [521, 421, 23]-code), using
- 1 times truncation [i] based on linear OA(2101, 522, F2, 23) (dual of [522, 421, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2100, 512, F2, 23) (dual of [512, 412, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(291, 512, F2, 21) (dual of [512, 421, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2101, 522, F2, 23) (dual of [522, 421, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2100, 521, F2, 22) (dual of [521, 421, 23]-code), using
- OOA 2-folding [i] based on linear OA(2100, 520, F2, 22) (dual of [520, 420, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2100, 260, F2, 2, 22) (dual of [(260, 2), 420, 23]-NRT-code), using
(78, 78+22, 2661)-Net in Base 2 — Upper bound on s
There is no (78, 100, 2662)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 272577 440785 974750 708858 673064 > 2100 [i]