Best Known (53, 53+19, s)-Nets in Base 2
(53, 53+19, 84)-Net over F2 — Constructive and digital
Digital (53, 72, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 24, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
(53, 53+19, 120)-Net over F2 — Digital
Digital (53, 72, 120)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(272, 120, F2, 2, 19) (dual of [(120, 2), 168, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(272, 132, F2, 2, 19) (dual of [(132, 2), 192, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(272, 264, F2, 19) (dual of [264, 192, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(269, 256, F2, 19) (dual of [256, 187, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(265, 256, F2, 17) (dual of [256, 191, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(257, 256, F2, 15) (dual of [256, 199, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 6, F2, 1) (dual of [6, 5, 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
- linear OA(21, 2, F2, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- Reed–Solomon code RS(1,2) [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(272, 264, F2, 19) (dual of [264, 192, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(272, 132, F2, 2, 19) (dual of [(132, 2), 192, 20]-NRT-code), using
(53, 53+19, 970)-Net in Base 2 — Upper bound on s
There is no (53, 72, 971)-net in base 2, because
- 1 times m-reduction [i] would yield (53, 71, 971)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2381 720300 356464 898516 > 271 [i]