Best Known (12, 12+13, s)-Nets in Base 27
(12, 12+13, 121)-Net over F27 — Constructive and digital
Digital (12, 25, 121)-net over F27, using
- net defined by OOA [i] based on linear OOA(2725, 121, F27, 13, 13) (dual of [(121, 13), 1548, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2725, 727, F27, 13) (dual of [727, 702, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2725, 727, F27, 13) (dual of [727, 702, 14]-code), using
(12, 12+13, 160)-Net in Base 27 — Constructive
(12, 25, 160)-net in base 27, using
- 3 times m-reduction [i] based on (12, 28, 160)-net in base 27, using
- base change [i] based on digital (5, 21, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- base change [i] based on digital (5, 21, 160)-net over F81, using
(12, 12+13, 337)-Net over F27 — Digital
Digital (12, 25, 337)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2725, 337, F27, 2, 13) (dual of [(337, 2), 649, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2725, 366, F27, 2, 13) (dual of [(366, 2), 707, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2725, 732, F27, 13) (dual of [732, 707, 14]-code), using
- construction XX applied to C1 = C([727,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([727,11]) [i] based on
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([727,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([727,11]) [i] based on
- OOA 2-folding [i] based on linear OA(2725, 732, F27, 13) (dual of [732, 707, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(2725, 366, F27, 2, 13) (dual of [(366, 2), 707, 14]-NRT-code), using
(12, 12+13, 61190)-Net in Base 27 — Upper bound on s
There is no (12, 25, 61191)-net in base 27, because
- 1 times m-reduction [i] would yield (12, 24, 61191)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 22529 376797 873393 887969 843861 962805 > 2724 [i]