Best Known (254−35, 254, s)-Nets in Base 2
(254−35, 254, 1062)-Net over F2 — Constructive and digital
Digital (219, 254, 1062)-net over F2, using
- 22 times duplication [i] based on digital (217, 252, 1062)-net over F2, using
- trace code for nets [i] based on digital (7, 42, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 42, 177)-net over F64, using
(254−35, 254, 3841)-Net over F2 — Digital
Digital (219, 254, 3841)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 3841, F2, 4, 35) (dual of [(3841, 4), 15110, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 4110, F2, 4, 35) (dual of [(4110, 4), 16186, 36]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2253, 4110, F2, 4, 35) (dual of [(4110, 4), 16187, 36]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2253, 16440, F2, 35) (dual of [16440, 16187, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2252, 16439, F2, 35) (dual of [16439, 16187, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(213, 55, F2, 5) (dual of [55, 42, 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
- 1 times code embedding in larger space [i] based on linear OA(2252, 16439, F2, 35) (dual of [16439, 16187, 36]-code), using
- OOA 4-folding [i] based on linear OA(2253, 16440, F2, 35) (dual of [16440, 16187, 36]-code), using
- 21 times duplication [i] based on linear OOA(2253, 4110, F2, 4, 35) (dual of [(4110, 4), 16187, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 4110, F2, 4, 35) (dual of [(4110, 4), 16186, 36]-NRT-code), using
(254−35, 254, 216735)-Net in Base 2 — Upper bound on s
There is no (219, 254, 216736)-net in base 2, because
- 1 times m-reduction [i] would yield (219, 253, 216736)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14474 744970 639322 239556 958473 657319 736655 444224 520732 824994 481541 758783 582027 > 2253 [i]