Best Known (78, 78+21, s)-Nets in Base 2
(78, 78+21, 152)-Net over F2 — Constructive and digital
Digital (78, 99, 152)-net over F2, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (3, 25, 38)-net over F16, using
(78, 78+21, 273)-Net over F2 — Digital
Digital (78, 99, 273)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(299, 273, F2, 2, 21) (dual of [(273, 2), 447, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(299, 546, F2, 21) (dual of [546, 447, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(299, 547, F2, 21) (dual of [547, 448, 22]-code), using
- adding a parity check bit [i] based on linear OA(298, 546, F2, 20) (dual of [546, 448, 21]-code), using
- construction XX applied to C1 = C([507,14]), C2 = C([1,16]), C3 = C1 + C2 = C([1,14]), 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(272, 511, F2, 16) (dual of [511, 439, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(263, 511, F2, 14) (dual of [511, 448, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 25, F2, 3) (dual of [25, 19, 4]-code or 25-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([1,16]), C3 = C1 + C2 = C([1,14]), and C∩ = C1 ∩ C2 = C([507,16]) [i] based on
- adding a parity check bit [i] based on linear OA(298, 546, F2, 20) (dual of [546, 448, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(299, 547, F2, 21) (dual of [547, 448, 22]-code), using
- OOA 2-folding [i] based on linear OA(299, 546, F2, 21) (dual of [546, 447, 22]-code), using
(78, 78+21, 4022)-Net in Base 2 — Upper bound on s
There is no (78, 99, 4023)-net in base 2, because
- 1 times m-reduction [i] would yield (78, 98, 4023)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 317203 003828 200785 375922 477226 > 298 [i]