Best Known (20, 20+13, s)-Nets in Base 27
(20, 20+13, 252)-Net over F27 — Constructive and digital
Digital (20, 33, 252)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 13, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
(20, 20+13, 264)-Net in Base 27 — Constructive
(20, 33, 264)-net in base 27, using
- 271 times duplication [i] based on (19, 32, 264)-net in base 27, using
- base change [i] based on digital (11, 24, 264)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (1, 14, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (0, 4, 82)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- base change [i] based on digital (11, 24, 264)-net over F81, using
(20, 20+13, 1763)-Net over F27 — Digital
Digital (20, 33, 1763)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2733, 1763, F27, 13) (dual of [1763, 1730, 14]-code), using
- 1023 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 24 times 0, 1, 82 times 0, 1, 187 times 0, 1, 303 times 0, 1, 417 times 0) [i] based on linear OA(2725, 732, F27, 13) (dual of [732, 707, 14]-code), using
- construction XX applied to C1 = C([727,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([727,11]) [i] based on
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([727,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([727,11]) [i] based on
- 1023 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 24 times 0, 1, 82 times 0, 1, 187 times 0, 1, 303 times 0, 1, 417 times 0) [i] based on linear OA(2725, 732, F27, 13) (dual of [732, 707, 14]-code), using
(20, 20+13, 4956653)-Net in Base 27 — Upper bound on s
There is no (20, 33, 4956654)-net in base 27, because
- 1 times m-reduction [i] would yield (20, 32, 4956654)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6362 688337 834606 962088 384177 926465 370684 522245 > 2732 [i]