Best Known (16, 16+17, s)-Nets in Base 27
(16, 16+17, 128)-Net over F27 — Constructive and digital
Digital (16, 33, 128)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 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 (4, 12, 64)-net over F27, using
(16, 16+17, 172)-Net in Base 27 — Constructive
(16, 33, 172)-net in base 27, using
- 3 times m-reduction [i] based on (16, 36, 172)-net in base 27, using
- base change [i] based on digital (7, 27, 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, 27, 172)-net over F81, using
(16, 16+17, 337)-Net over F27 — Digital
Digital (16, 33, 337)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2733, 337, F27, 2, 17) (dual of [(337, 2), 641, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2733, 366, F27, 2, 17) (dual of [(366, 2), 699, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2733, 732, F27, 17) (dual of [732, 699, 18]-code), using
- construction XX applied to C1 = C([727,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([727,15]) [i] based on
- linear OA(2731, 728, F27, 16) (dual of [728, 697, 17]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [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(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(2729, 728, F27, 15) (dual of [728, 699, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [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([727,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([727,15]) [i] based on
- OOA 2-folding [i] based on linear OA(2733, 732, F27, 17) (dual of [732, 699, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(2733, 366, F27, 2, 17) (dual of [(366, 2), 699, 18]-NRT-code), using
(16, 16+17, 76939)-Net in Base 27 — Upper bound on s
There is no (16, 33, 76940)-net in base 27, because
- 1 times m-reduction [i] would yield (16, 32, 76940)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6362 908124 916560 761557 535080 172656 048018 504129 > 2732 [i]