Best Known (45, 45+37, s)-Nets in Base 27
(45, 45+37, 210)-Net over F27 — Constructive and digital
Digital (45, 82, 210)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 16, 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, 22, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (7, 44, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (4, 16, 64)-net over F27, using
(45, 45+37, 370)-Net in Base 27 — Constructive
(45, 82, 370)-net in base 27, using
- t-expansion [i] based on (43, 82, 370)-net in base 27, using
- 26 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 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, 81, 370)-net over F81, using
- 26 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(45, 45+37, 1019)-Net over F27 — Digital
Digital (45, 82, 1019)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2782, 1019, F27, 37) (dual of [1019, 937, 38]-code), using
- 275 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 32 times 0, 1, 51 times 0, 1, 70 times 0, 1, 84 times 0) [i] based on linear OA(2770, 732, F27, 37) (dual of [732, 662, 38]-code), using
- construction XX applied to C1 = C([727,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([727,35]) [i] based on
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [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([727,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([727,35]) [i] based on
- 275 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 32 times 0, 1, 51 times 0, 1, 70 times 0, 1, 84 times 0) [i] based on linear OA(2770, 732, F27, 37) (dual of [732, 662, 38]-code), using
(45, 45+37, 802211)-Net in Base 27 — Upper bound on s
There is no (45, 82, 802212)-net in base 27, because
- 1 times m-reduction [i] would yield (45, 81, 802212)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 87 191388 401522 381219 081879 955768 342890 247938 212267 982217 267245 416434 774508 385564 675047 061545 437806 369617 751701 119225 > 2781 [i]