Best Known (42, 42+35, s)-Nets in Base 27
(42, 42+35, 204)-Net over F27 — Constructive and digital
Digital (42, 77, 204)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 15, 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, 21, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (6, 41, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 15, 64)-net over F27, using
(42, 42+35, 370)-Net in Base 27 — Constructive
(42, 77, 370)-net in base 27, using
- 27 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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, 78, 370)-net over F81, using
(42, 42+35, 928)-Net over F27 — Digital
Digital (42, 77, 928)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2777, 928, F27, 35) (dual of [928, 851, 36]-code), using
- 185 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 29 times 0, 1, 50 times 0, 1, 71 times 0) [i] based on linear OA(2766, 732, F27, 35) (dual of [732, 666, 36]-code), using
- construction XX applied to C1 = C([727,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([727,33]) [i] based on
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [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([727,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([727,33]) [i] based on
- 185 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 29 times 0, 1, 50 times 0, 1, 71 times 0) [i] based on linear OA(2766, 732, F27, 35) (dual of [732, 666, 36]-code), using
(42, 42+35, 691838)-Net in Base 27 — Upper bound on s
There is no (42, 77, 691839)-net in base 27, because
- 1 times m-reduction [i] would yield (42, 76, 691839)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6 076541 005941 175002 494735 050899 276689 077696 362101 776936 721327 883276 233050 889194 972799 347052 071654 584594 675527 > 2776 [i]