Best Known (8, 8+8, s)-Nets in Base 27
(8, 8+8, 183)-Net over F27 — Constructive and digital
Digital (8, 16, 183)-net over F27, using
- 271 times duplication [i] based on digital (7, 15, 183)-net over F27, using
- net defined by OOA [i] based on linear OOA(2715, 183, F27, 8, 8) (dual of [(183, 8), 1449, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
- net defined by OOA [i] based on linear OOA(2715, 183, F27, 8, 8) (dual of [(183, 8), 1449, 9]-NRT-code), using
(8, 8+8, 434)-Net over F27 — Digital
Digital (8, 16, 434)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2716, 434, F27, 8) (dual of [434, 418, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2716, 728, F27, 8) (dual of [728, 712, 9]-code), using
(8, 8+8, 45239)-Net in Base 27 — Upper bound on s
There is no (8, 16, 45240)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 79769 347496 712690 880961 > 2716 [i]