Best Known (79−21, 79, s)-Nets in Base 2
(79−21, 79, 84)-Net over F2 — Constructive and digital
Digital (58, 79, 84)-net over F2, using
- 21 times duplication [i] based on digital (57, 78, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 26, 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, 26, 28)-net over F8, using
(79−21, 79, 123)-Net over F2 — Digital
Digital (58, 79, 123)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(279, 123, F2, 2, 21) (dual of [(123, 2), 167, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(279, 134, F2, 2, 21) (dual of [(134, 2), 189, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(279, 268, F2, 21) (dual of [268, 189, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 269, F2, 21) (dual of [269, 190, 22]-code), using
- construction XX applied to C1 = C([253,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), 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(269, 255, F2, 19) (dual of [255, 186, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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(265, 255, F2, 17) (dual of [255, 190, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([253,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(279, 269, F2, 21) (dual of [269, 190, 22]-code), using
- OOA 2-folding [i] based on linear OA(279, 268, F2, 21) (dual of [268, 189, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(279, 134, F2, 2, 21) (dual of [(134, 2), 189, 22]-NRT-code), using
(79−21, 79, 994)-Net in Base 2 — Upper bound on s
There is no (58, 79, 995)-net in base 2, because
- 1 times m-reduction [i] would yield (58, 78, 995)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 302623 065348 156521 685924 > 278 [i]