Best Known (25, 25+26, s)-Nets in Base 27
(25, 25+26, 152)-Net over F27 — Constructive and digital
Digital (25, 51, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 19, 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 (6, 32, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 19, 76)-net over F27, using
(25, 25+26, 172)-Net in Base 27 — Constructive
(25, 51, 172)-net in base 27, using
- 21 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(25, 25+26, 366)-Net over F27 — Digital
Digital (25, 51, 366)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2751, 366, F27, 2, 26) (dual of [(366, 2), 681, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2751, 732, F27, 26) (dual of [732, 681, 27]-code), using
- construction XX applied to C1 = C([727,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([727,24]) [i] based on
- linear OA(2749, 728, F27, 25) (dual of [728, 679, 26]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2749, 728, F27, 25) (dual of [728, 679, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2751, 728, F27, 26) (dual of [728, 677, 27]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [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([727,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([727,24]) [i] based on
- OOA 2-folding [i] based on linear OA(2751, 732, F27, 26) (dual of [732, 681, 27]-code), using
(25, 25+26, 89898)-Net in Base 27 — Upper bound on s
There is no (25, 51, 89899)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 9 990669 416827 874899 527704 479250 196848 476066 758869 201580 084294 069792 868263 > 2751 [i]