Best Known (31, 31+25, s)-Nets in Base 27
(31, 31+25, 182)-Net over F27 — Constructive and digital
Digital (31, 56, 182)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 21, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (10, 35, 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 (9, 21, 88)-net over F27, using
(31, 31+25, 370)-Net in Base 27 — Constructive
(31, 56, 370)-net in base 27, using
- 4 times m-reduction [i] based on (31, 60, 370)-net in base 27, using
- base change [i] based on digital (16, 45, 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, 45, 370)-net over F81, using
(31, 31+25, 851)-Net over F27 — Digital
Digital (31, 56, 851)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2756, 851, F27, 25) (dual of [851, 795, 26]-code), using
- 110 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 66 times 0) [i] based on linear OA(2750, 735, F27, 25) (dual of [735, 685, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(2749, 730, F27, 25) (dual of [730, 681, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2745, 730, F27, 23) (dual of [730, 685, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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 C([0,12]) ⊂ C([0,11]) [i] based on
- 110 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 66 times 0) [i] based on linear OA(2750, 735, F27, 25) (dual of [735, 685, 26]-code), using
(31, 31+25, 739267)-Net in Base 27 — Upper bound on s
There is no (31, 56, 739268)-net in base 27, because
- 1 times m-reduction [i] would yield (31, 55, 739268)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 5 308943 827083 008003 055795 507880 688599 840914 876535 389872 622565 351931 863476 371457 > 2755 [i]