Best Known (126−23, 126, s)-Nets in Base 2
(126−23, 126, 260)-Net over F2 — Constructive and digital
Digital (103, 126, 260)-net over F2, using
- t-expansion [i] based on digital (102, 126, 260)-net over F2, using
- 2 times m-reduction [i] based on 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
- 2 times m-reduction [i] based on digital (102, 128, 260)-net over F2, using
(126−23, 126, 678)-Net over F2 — Digital
Digital (103, 126, 678)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2126, 678, F2, 3, 23) (dual of [(678, 3), 1908, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 687, F2, 3, 23) (dual of [(687, 3), 1935, 24]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2124, 687, F2, 3, 23) (dual of [(687, 3), 1937, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2124, 2061, F2, 23) (dual of [2061, 1937, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2123, 2060, F2, 23) (dual of [2060, 1937, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2123, 2060, F2, 23) (dual of [2060, 1937, 24]-code), using
- OOA 3-folding [i] based on linear OA(2124, 2061, F2, 23) (dual of [2061, 1937, 24]-code), using
- 22 times duplication [i] based on linear OOA(2124, 687, F2, 3, 23) (dual of [(687, 3), 1937, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 687, F2, 3, 23) (dual of [(687, 3), 1935, 24]-NRT-code), using
(126−23, 126, 12920)-Net in Base 2 — Upper bound on s
There is no (103, 126, 12921)-net in base 2, because
- 1 times m-reduction [i] would yield (103, 125, 12921)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 42 560856 704543 216323 573588 808648 623032 > 2125 [i]