Best Known (233−42, 233, s)-Nets in Base 2
(233−42, 233, 320)-Net over F2 — Constructive and digital
Digital (191, 233, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (191, 235, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 47, 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, 47, 64)-net over F32, using
(233−42, 233, 879)-Net over F2 — Digital
Digital (191, 233, 879)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 879, F2, 2, 42) (dual of [(879, 2), 1525, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2233, 1030, F2, 2, 42) (dual of [(1030, 2), 1827, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2233, 2060, F2, 42) (dual of [2060, 1827, 43]-code), using
- strength reduction [i] based on linear OA(2233, 2060, F2, 43) (dual of [2060, 1827, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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(42) ⊂ Ce(40) [i] based on
- strength reduction [i] based on linear OA(2233, 2060, F2, 43) (dual of [2060, 1827, 44]-code), using
- OOA 2-folding [i] based on linear OA(2233, 2060, F2, 42) (dual of [2060, 1827, 43]-code), using
- discarding factors / shortening the dual code based on linear OOA(2233, 1030, F2, 2, 42) (dual of [(1030, 2), 1827, 43]-NRT-code), using
(233−42, 233, 18957)-Net in Base 2 — Upper bound on s
There is no (191, 233, 18958)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 13809 851120 306831 215248 526534 610301 128598 144909 255786 932573 296294 167072 > 2233 [i]