Best Known (116, 116+26, s)-Nets in Base 2
(116, 116+26, 265)-Net over F2 — Constructive and digital
Digital (116, 142, 265)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 5)-net over F2, using
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 1 and N(F) ≥ 5, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- digital (102, 128, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 32, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 32, 65)-net over F16, using
- digital (1, 14, 5)-net over F2, using
(116, 116+26, 544)-Net over F2 — Digital
Digital (116, 142, 544)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2142, 544, F2, 26) (dual of [544, 402, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2142, 1062, F2, 26) (dual of [1062, 920, 27]-code), using
- construction X applied to C([1,26]) ⊂ C([1,20]) [i] based on
- linear OA(2130, 1023, F2, 26) (dual of [1023, 893, 27]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2100, 1023, F2, 20) (dual of [1023, 923, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(212, 39, F2, 5) (dual of [39, 27, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(211, 32, F2, 5) (dual of [32, 21, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 31 = 25−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 31 = 25−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(26, 7, F2, 5) (dual of [7, 1, 6]-code), using
- strength reduction [i] based on linear OA(26, 7, F2, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,2)), using
- dual of repetition code with length 7 [i]
- strength reduction [i] based on linear OA(26, 7, F2, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,2)), using
- linear OA(21, 7, F2, 1) (dual of [7, 6, 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 X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- construction X applied to C([1,26]) ⊂ C([1,20]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2142, 1062, F2, 26) (dual of [1062, 920, 27]-code), using
(116, 116+26, 10985)-Net in Base 2 — Upper bound on s
There is no (116, 142, 10986)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5 576769 593209 744484 728076 393511 117410 038584 > 2142 [i]