Best Known (53, 71, s)-Nets in Base 2
(53, 71, 84)-Net over F2 — Constructive and digital
Digital (53, 71, 84)-net over F2, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (5, 24, 28)-net over F8, using
(53, 71, 131)-Net over F2 — Digital
Digital (53, 71, 131)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 131, F2, 2, 18) (dual of [(131, 2), 191, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(271, 131, F2, 2, 19) (dual of [(131, 2), 191, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(271, 262, F2, 19) (dual of [262, 191, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(270, 261, F2, 19) (dual of [261, 191, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(21, 5, F2, 1) (dual of [5, 4, 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(18) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(270, 261, F2, 19) (dual of [261, 191, 20]-code), using
- OOA 2-folding [i] based on linear OA(271, 262, F2, 19) (dual of [262, 191, 20]-code), using
- strength reduction [i] based on linear OOA(271, 131, F2, 2, 19) (dual of [(131, 2), 191, 20]-NRT-code), using
(53, 71, 970)-Net in Base 2 — Upper bound on s
There is no (53, 71, 971)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2381 720300 356464 898516 > 271 [i]