Best Known (122, 122+26, s)-Nets in Base 2
(122, 122+26, 320)-Net over F2 — Constructive and digital
Digital (122, 148, 320)-net over F2, using
- 23 times duplication [i] based on digital (119, 145, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 29, 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, 29, 64)-net over F32, using
(122, 122+26, 735)-Net over F2 — Digital
Digital (122, 148, 735)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2148, 735, F2, 2, 26) (dual of [(735, 2), 1322, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2148, 1031, F2, 2, 26) (dual of [(1031, 2), 1914, 27]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2147, 1031, F2, 2, 26) (dual of [(1031, 2), 1915, 27]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2145, 1030, F2, 2, 26) (dual of [(1030, 2), 1915, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2145, 2060, F2, 26) (dual of [2060, 1915, 27]-code), using
- strength reduction [i] based on linear OA(2145, 2060, F2, 27) (dual of [2060, 1915, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(26) ⊂ Ce(24) [i] based on
- strength reduction [i] based on linear OA(2145, 2060, F2, 27) (dual of [2060, 1915, 28]-code), using
- OOA 2-folding [i] based on linear OA(2145, 2060, F2, 26) (dual of [2060, 1915, 27]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2145, 1030, F2, 2, 26) (dual of [(1030, 2), 1915, 27]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2147, 1031, F2, 2, 26) (dual of [(1031, 2), 1915, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2148, 1031, F2, 2, 26) (dual of [(1031, 2), 1914, 27]-NRT-code), using
(122, 122+26, 15134)-Net in Base 2 — Upper bound on s
There is no (122, 148, 15135)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 356 919566 534438 235611 290004 370958 711967 373828 > 2148 [i]