Best Known (63, 80, s)-Nets in Base 2
(63, 80, 152)-Net over F2 — Constructive and digital
Digital (63, 80, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 20, 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
(63, 80, 268)-Net over F2 — Digital
Digital (63, 80, 268)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(280, 268, F2, 2, 17) (dual of [(268, 2), 456, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(280, 272, F2, 2, 17) (dual of [(272, 2), 464, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(280, 544, F2, 17) (dual of [544, 464, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 545, F2, 17) (dual of [545, 465, 18]-code), using
- construction XX applied to C1 = C([507,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([507,12]) [i] based on
- linear OA(264, 511, F2, 15) (dual of [511, 447, 16]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,10}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(255, 511, F2, 13) (dual of [511, 456, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,12}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(246, 511, F2, 11) (dual of [511, 465, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([507,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([507,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(280, 545, F2, 17) (dual of [545, 465, 18]-code), using
- OOA 2-folding [i] based on linear OA(280, 544, F2, 17) (dual of [544, 464, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(280, 272, F2, 2, 17) (dual of [(272, 2), 464, 18]-NRT-code), using
(63, 80, 3523)-Net in Base 2 — Upper bound on s
There is no (63, 80, 3524)-net in base 2, because
- 1 times m-reduction [i] would yield (63, 79, 3524)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 605438 368944 965762 722408 > 279 [i]