Best Known (27, 27+31, s)-Nets in Base 27
(27, 27+31, 152)-Net over F27 — Constructive and digital
Digital (27, 58, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 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, 37, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 21, 76)-net over F27, using
(27, 27+31, 172)-Net in Base 27 — Constructive
(27, 58, 172)-net in base 27, using
- 22 times m-reduction [i] based on (27, 80, 172)-net in base 27, using
- base change [i] based on digital (7, 60, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 60, 172)-net over F81, using
(27, 27+31, 303)-Net over F27 — Digital
Digital (27, 58, 303)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2758, 303, F27, 2, 31) (dual of [(303, 2), 548, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2758, 366, F27, 2, 31) (dual of [(366, 2), 674, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2758, 732, F27, 31) (dual of [732, 674, 32]-code), using
- construction XX applied to C1 = C([727,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([727,29]) [i] based on
- linear OA(2756, 728, F27, 30) (dual of [728, 672, 31]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [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(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(2754, 728, F27, 29) (dual of [728, 674, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [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([727,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([727,29]) [i] based on
- OOA 2-folding [i] based on linear OA(2758, 732, F27, 31) (dual of [732, 674, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2758, 366, F27, 2, 31) (dual of [(366, 2), 674, 32]-NRT-code), using
(27, 27+31, 67908)-Net in Base 27 — Upper bound on s
There is no (27, 58, 67909)-net in base 27, because
- 1 times m-reduction [i] would yield (27, 57, 67909)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3870 235762 842529 869397 988318 845047 042338 261268 676867 319617 632337 951229 617844 220675 > 2757 [i]