Best Known (113−20, 113, s)-Nets in Base 2
(113−20, 113, 260)-Net over F2 — Constructive and digital
Digital (93, 113, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (93, 116, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 29, 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, 29, 65)-net over F16, using
(113−20, 113, 687)-Net over F2 — Digital
Digital (93, 113, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2113, 687, F2, 3, 20) (dual of [(687, 3), 1948, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2113, 2061, F2, 20) (dual of [2061, 1948, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2111, 2059, F2, 20) (dual of [2059, 1948, 21]-code), using
- 1 times truncation [i] based on linear OA(2112, 2060, F2, 21) (dual of [2060, 1948, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2111, 2048, F2, 21) (dual of [2048, 1937, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2100, 2048, F2, 19) (dual of [2048, 1948, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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 X applied to Ce(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2112, 2060, F2, 21) (dual of [2060, 1948, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2111, 2059, F2, 20) (dual of [2059, 1948, 21]-code), using
- OOA 3-folding [i] based on linear OA(2113, 2061, F2, 20) (dual of [2061, 1948, 21]-code), using
(113−20, 113, 11404)-Net in Base 2 — Upper bound on s
There is no (93, 113, 11405)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10392 133640 100213 568426 354471 763934 > 2113 [i]