Best Known (155−27, 155, s)-Nets in Base 2
(155−27, 155, 320)-Net over F2 — Constructive and digital
Digital (128, 155, 320)-net over F2, using
- t-expansion [i] based on digital (127, 155, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 31, 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, 31, 64)-net over F32, using
(155−27, 155, 779)-Net over F2 — Digital
Digital (128, 155, 779)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 779, F2, 2, 27) (dual of [(779, 2), 1403, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 1042, F2, 2, 27) (dual of [(1042, 2), 1929, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2155, 2084, F2, 27) (dual of [2084, 1929, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 2085, F2, 27) (dual of [2085, 1930, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(20) [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(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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 5, F2, 1) (dual of [5, 4, 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 XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2155, 2085, F2, 27) (dual of [2085, 1930, 28]-code), using
- OOA 2-folding [i] based on linear OA(2155, 2084, F2, 27) (dual of [2084, 1929, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 1042, F2, 2, 27) (dual of [(1042, 2), 1929, 28]-NRT-code), using
(155−27, 155, 20847)-Net in Base 2 — Upper bound on s
There is no (128, 155, 20848)-net in base 2, because
- 1 times m-reduction [i] would yield (128, 154, 20848)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 22840 088365 987274 762339 342937 500496 136671 627003 > 2154 [i]