Best Known (238−35, 238, s)-Nets in Base 2
(238−35, 238, 624)-Net over F2 — Constructive and digital
Digital (203, 238, 624)-net over F2, using
- 2 times m-reduction [i] based on digital (203, 240, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
(238−35, 238, 2332)-Net over F2 — Digital
Digital (203, 238, 2332)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2238, 2332, F2, 3, 35) (dual of [(2332, 3), 6758, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2238, 2749, F2, 3, 35) (dual of [(2749, 3), 8009, 36]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2235, 2748, F2, 3, 35) (dual of [(2748, 3), 8009, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2235, 8244, F2, 35) (dual of [8244, 8009, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(213, 52, F2, 5) (dual of [52, 39, 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(34) ⊂ Ce(28) [i] based on
- OOA 3-folding [i] based on linear OA(2235, 8244, F2, 35) (dual of [8244, 8009, 36]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2235, 2748, F2, 3, 35) (dual of [(2748, 3), 8009, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2238, 2749, F2, 3, 35) (dual of [(2749, 3), 8009, 36]-NRT-code), using
(238−35, 238, 112865)-Net in Base 2 — Upper bound on s
There is no (203, 238, 112866)-net in base 2, because
- 1 times m-reduction [i] would yield (203, 237, 112866)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 220871 541256 152417 553285 461949 298288 906085 245925 435693 738621 523929 005902 > 2237 [i]