Best Known (71, 71+17, s)-Nets in Base 2
(71, 71+17, 196)-Net over F2 — Constructive and digital
Digital (71, 88, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 22, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
(71, 71+17, 408)-Net over F2 — Digital
Digital (71, 88, 408)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(288, 408, F2, 2, 17) (dual of [(408, 2), 728, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(288, 530, F2, 2, 17) (dual of [(530, 2), 972, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(288, 1060, F2, 17) (dual of [1060, 972, 18]-code), using
- construction XX applied to C1 = C([1019,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1019,12]) [i] based on
- linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,10}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(281, 1023, F2, 17) (dual of [1023, 942, 18]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,12}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(251, 1023, F2, 11) (dual of [1023, 972, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-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, 11, F2, 1) (dual of [11, 10, 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([1019,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1019,12]) [i] based on
- OOA 2-folding [i] based on linear OA(288, 1060, F2, 17) (dual of [1060, 972, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(288, 530, F2, 2, 17) (dual of [(530, 2), 972, 18]-NRT-code), using
(71, 71+17, 7058)-Net in Base 2 — Upper bound on s
There is no (71, 88, 7059)-net in base 2, because
- 1 times m-reduction [i] would yield (71, 87, 7059)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 154 908917 234600 224723 105534 > 287 [i]