Best Known (17, 35, s)-Nets in Base 27
(17, 35, 128)-Net over F27 — Constructive and digital
Digital (17, 35, 128)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 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 (4, 13, 64)-net over F27, using
(17, 35, 172)-Net in Base 27 — Constructive
(17, 35, 172)-net in base 27, using
- 5 times m-reduction [i] based on (17, 40, 172)-net in base 27, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
(17, 35, 341)-Net over F27 — Digital
Digital (17, 35, 341)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2735, 341, F27, 2, 18) (dual of [(341, 2), 647, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2735, 366, F27, 2, 18) (dual of [(366, 2), 697, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2735, 732, F27, 18) (dual of [732, 697, 19]-code), using
- construction XX applied to C1 = C([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- linear OA(2733, 728, F27, 17) (dual of [728, 695, 18]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2733, 728, F27, 17) (dual of [728, 695, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2731, 728, F27, 16) (dual of [728, 697, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- OOA 2-folding [i] based on linear OA(2735, 732, F27, 18) (dual of [732, 697, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(2735, 366, F27, 2, 18) (dual of [(366, 2), 697, 19]-NRT-code), using
(17, 35, 58770)-Net in Base 27 — Upper bound on s
There is no (17, 35, 58771)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 125 243112 915898 276002 214517 170558 508008 460232 515551 > 2735 [i]