Best Known (113, 113+24, s)-Nets in Base 2
(113, 113+24, 320)-Net over F2 — Constructive and digital
Digital (113, 137, 320)-net over F2, using
- 22 times duplication [i] based on digital (111, 135, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 27, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 27, 64)-net over F32, using
(113, 113+24, 718)-Net over F2 — Digital
Digital (113, 137, 718)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 718, F2, 2, 24) (dual of [(718, 2), 1299, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 1032, F2, 2, 24) (dual of [(1032, 2), 1927, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2137, 2064, F2, 24) (dual of [2064, 1927, 25]-code), using
- 1 times truncation [i] based on linear OA(2138, 2065, F2, 25) (dual of [2065, 1927, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2133, 2049, F2, 25) (dual of [2049, 1916, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25, 16, F2, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 1 times truncation [i] based on linear OA(2138, 2065, F2, 25) (dual of [2065, 1927, 26]-code), using
- OOA 2-folding [i] based on linear OA(2137, 2064, F2, 24) (dual of [2064, 1927, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 1032, F2, 2, 24) (dual of [(1032, 2), 1927, 25]-NRT-code), using
(113, 113+24, 14440)-Net in Base 2 — Upper bound on s
There is no (113, 137, 14441)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 174236 232116 943851 185763 756769 821928 688296 > 2137 [i]