Best Known (67−30, 67, s)-Nets in Base 27
(67−30, 67, 192)-Net over F27 — Constructive and digital
Digital (37, 67, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 26, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (11, 41, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 26, 96)-net over F27, using
(67−30, 67, 370)-Net in Base 27 — Constructive
(37, 67, 370)-net in base 27, using
- 17 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
(67−30, 67, 932)-Net over F27 — Digital
Digital (37, 67, 932)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2767, 932, F27, 30) (dual of [932, 865, 31]-code), using
- 190 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 29 times 0, 1, 51 times 0, 1, 77 times 0) [i] based on linear OA(2756, 731, F27, 30) (dual of [731, 675, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(2756, 729, F27, 30) (dual of [729, 673, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- 190 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 29 times 0, 1, 51 times 0, 1, 77 times 0) [i] based on linear OA(2756, 731, F27, 30) (dual of [731, 675, 31]-code), using
(67−30, 67, 611241)-Net in Base 27 — Upper bound on s
There is no (37, 67, 611242)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 796854 778321 592829 752411 911259 205283 861843 473071 983081 821066 498818 324465 431117 995915 710354 233193 > 2767 [i]