Best Known (28, 28+32, s)-Nets in Base 27
(28, 28+32, 152)-Net over F27 — Constructive and digital
Digital (28, 60, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 22, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (6, 38, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 22, 76)-net over F27, using
(28, 28+32, 224)-Net in Base 27 — Constructive
(28, 60, 224)-net in base 27, using
- base change [i] based on digital (13, 45, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
(28, 28+32, 313)-Net over F27 — Digital
Digital (28, 60, 313)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2760, 313, F27, 2, 32) (dual of [(313, 2), 566, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2760, 366, F27, 2, 32) (dual of [(366, 2), 672, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2760, 732, F27, 32) (dual of [732, 672, 33]-code), using
- construction XX applied to C1 = C([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2756, 728, F27, 30) (dual of [728, 672, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- OOA 2-folding [i] based on linear OA(2760, 732, F27, 32) (dual of [732, 672, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(2760, 366, F27, 2, 32) (dual of [(366, 2), 672, 33]-NRT-code), using
(28, 28+32, 60970)-Net in Base 27 — Upper bound on s
There is no (28, 60, 60971)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 76 181910 923117 860179 665725 762018 866439 649178 088787 854350 028768 827953 956900 806740 364897 > 2760 [i]