Best Known (51, 51+31, s)-Nets in Base 27
(51, 51+31, 274)-Net over F27 — Constructive and digital
Digital (51, 82, 274)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 11, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (4, 14, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (4, 19, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (7, 38, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (4, 11, 64)-net over F27, using
(51, 51+31, 437)-Net in Base 27 — Constructive
(51, 82, 437)-net in base 27, using
- net defined by OOA [i] based on OOA(2782, 437, S27, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(2782, 6556, S27, 31), using
- discarding factors based on OA(2782, 6563, S27, 31), using
- discarding parts of the base [i] based on linear OA(8161, 6563, F81, 31) (dual of [6563, 6502, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- discarding parts of the base [i] based on linear OA(8161, 6563, F81, 31) (dual of [6563, 6502, 32]-code), using
- discarding factors based on OA(2782, 6563, S27, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(2782, 6556, S27, 31), using
(51, 51+31, 3801)-Net over F27 — Digital
Digital (51, 82, 3801)-net over F27, using
(51, 51+31, large)-Net in Base 27 — Upper bound on s
There is no (51, 82, large)-net in base 27, because
- 29 times m-reduction [i] would yield (51, 53, large)-net in base 27, but