Best Known (259−45, 259, s)-Nets in Base 2
(259−45, 259, 380)-Net over F2 — Constructive and digital
Digital (214, 259, 380)-net over F2, using
- t-expansion [i] based on digital (213, 259, 380)-net over F2, using
- 1 times m-reduction [i] based on digital (213, 260, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 52, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 52, 76)-net over F32, using
- 1 times m-reduction [i] based on digital (213, 260, 380)-net over F2, using
(259−45, 259, 1048)-Net over F2 — Digital
Digital (214, 259, 1048)-net over F2, using
- 21 times duplication [i] based on digital (213, 258, 1048)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 1048, F2, 2, 45) (dual of [(1048, 2), 1838, 46]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2256, 1047, F2, 2, 45) (dual of [(1047, 2), 1838, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2256, 2094, F2, 45) (dual of [2094, 1838, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(38) [i] based on
- linear OA(2243, 2048, F2, 45) (dual of [2048, 1805, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(44) ⊂ Ce(38) [i] based on
- OOA 2-folding [i] based on linear OA(2256, 2094, F2, 45) (dual of [2094, 1838, 46]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2256, 1047, F2, 2, 45) (dual of [(1047, 2), 1838, 46]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 1048, F2, 2, 45) (dual of [(1048, 2), 1838, 46]-NRT-code), using
(259−45, 259, 30665)-Net in Base 2 — Upper bound on s
There is no (214, 259, 30666)-net in base 2, because
- 1 times m-reduction [i] would yield (214, 258, 30666)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463239 880052 785358 897855 192731 197540 017429 848368 745957 487907 493094 888127 372368 > 2258 [i]