Best Known (62−28, 62, s)-Nets in Base 27
(62−28, 62, 188)-Net over F27 — Constructive and digital
Digital (34, 62, 188)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 24, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (10, 38, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27 (see above)
- digital (10, 24, 94)-net over F27, using
(62−28, 62, 370)-Net in Base 27 — Constructive
(34, 62, 370)-net in base 27, using
- 10 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
(62−28, 62, 847)-Net over F27 — Digital
Digital (34, 62, 847)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2762, 847, F27, 28) (dual of [847, 785, 29]-code), using
- 107 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 29 times 0, 1, 55 times 0) [i] based on linear OA(2754, 732, F27, 28) (dual of [732, 678, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(2753, 729, F27, 28) (dual of [729, 676, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2751, 729, F27, 26) (dual of [729, 678, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(271, 3, F27, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- 107 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 29 times 0, 1, 55 times 0) [i] based on linear OA(2754, 732, F27, 28) (dual of [732, 678, 29]-code), using
(62−28, 62, 507428)-Net in Base 27 — Upper bound on s
There is no (34, 62, 507429)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 55533 728094 376977 619306 478409 427196 605749 476668 910994 036244 490709 830309 308263 199074 853073 > 2762 [i]