Best Known (59, 59+21, s)-Nets in Base 2
(59, 59+21, 84)-Net over F2 — Constructive and digital
Digital (59, 80, 84)-net over F2, using
- 1 times m-reduction [i] based on digital (59, 81, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 27, 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, 27, 28)-net over F8, using
(59, 59+21, 129)-Net over F2 — Digital
Digital (59, 80, 129)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(280, 129, F2, 2, 21) (dual of [(129, 2), 178, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(280, 135, F2, 2, 21) (dual of [(135, 2), 190, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(280, 270, F2, 21) (dual of [270, 190, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 271, F2, 21) (dual of [271, 191, 22]-code), using
- adding a parity check bit [i] based on linear OA(279, 270, F2, 20) (dual of [270, 191, 21]-code), using
- construction XX applied to C1 = C([253,16]), C2 = C([1,18]), C3 = C1 + C2 = C([1,16]), and C∩ = C1 ∩ C2 = C([253,18]) [i] based on
- linear OA(273, 255, F2, 19) (dual of [255, 182, 20]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(268, 255, F2, 18) (dual of [255, 187, 19]-code), using 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(277, 255, F2, 21) (dual of [255, 178, 22]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(264, 255, F2, 16) (dual of [255, 191, 17]-code), using 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, 10, F2, 1) (dual of [10, 9, 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, 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 (see above)
- construction XX applied to C1 = C([253,16]), C2 = C([1,18]), C3 = C1 + C2 = C([1,16]), and C∩ = C1 ∩ C2 = C([253,18]) [i] based on
- adding a parity check bit [i] based on linear OA(279, 270, F2, 20) (dual of [270, 191, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 271, F2, 21) (dual of [271, 191, 22]-code), using
- OOA 2-folding [i] based on linear OA(280, 270, F2, 21) (dual of [270, 190, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(280, 135, F2, 2, 21) (dual of [(135, 2), 190, 22]-NRT-code), using
(59, 59+21, 1067)-Net in Base 2 — Upper bound on s
There is no (59, 80, 1068)-net in base 2, because
- 1 times m-reduction [i] would yield (59, 79, 1068)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 608372 490229 812748 010319 > 279 [i]