Best Known (137, 137+29, s)-Nets in Base 2
(137, 137+29, 320)-Net over F2 — Constructive and digital
Digital (137, 166, 320)-net over F2, using
- 21 times duplication [i] based on digital (136, 165, 320)-net over F2, using
- t-expansion [i] based on digital (135, 165, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 33, 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, 33, 64)-net over F32, using
- t-expansion [i] based on digital (135, 165, 320)-net over F2, using
(137, 137+29, 799)-Net over F2 — Digital
Digital (137, 166, 799)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2166, 799, F2, 2, 29) (dual of [(799, 2), 1432, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 1042, F2, 2, 29) (dual of [(1042, 2), 1918, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2166, 2084, F2, 29) (dual of [2084, 1918, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2166, 2085, F2, 29) (dual of [2085, 1919, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(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(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(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2166, 2085, F2, 29) (dual of [2085, 1919, 30]-code), using
- OOA 2-folding [i] based on linear OA(2166, 2084, F2, 29) (dual of [2084, 1918, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 1042, F2, 2, 29) (dual of [(1042, 2), 1918, 30]-NRT-code), using
(137, 137+29, 21325)-Net in Base 2 — Upper bound on s
There is no (137, 166, 21326)-net in base 2, because
- 1 times m-reduction [i] would yield (137, 165, 21326)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 46 791029 463025 405187 553789 007366 766807 620063 972824 > 2165 [i]