Best Known (77, 77+21, s)-Nets in Base 2
(77, 77+21, 152)-Net over F2 — Constructive and digital
Digital (77, 98, 152)-net over F2, using
- 22 times duplication [i] based on digital (75, 96, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 24, 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
- trace code for nets [i] based on digital (3, 24, 38)-net over F16, using
(77, 77+21, 272)-Net over F2 — Digital
Digital (77, 98, 272)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(298, 272, F2, 2, 21) (dual of [(272, 2), 446, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(298, 544, F2, 21) (dual of [544, 446, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(298, 545, F2, 21) (dual of [545, 447, 22]-code), using
- construction XX applied to C1 = C([507,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([507,16]) [i] based on
- linear OA(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,14}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(291, 511, F2, 21) (dual of [511, 420, 22]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,16}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(264, 511, F2, 15) (dual of [511, 447, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(26, 24, F2, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- 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 XX applied to C1 = C([507,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([507,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(298, 545, F2, 21) (dual of [545, 447, 22]-code), using
- OOA 2-folding [i] based on linear OA(298, 544, F2, 21) (dual of [544, 446, 22]-code), using
(77, 77+21, 3752)-Net in Base 2 — Upper bound on s
There is no (77, 98, 3753)-net in base 2, because
- 1 times m-reduction [i] would yield (77, 97, 3753)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 158759 906927 295501 548051 147284 > 297 [i]