Best Known (49−25, 49, s)-Nets in Base 27
(49−25, 49, 152)-Net over F27 — Constructive and digital
Digital (24, 49, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 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, 31, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 18, 76)-net over F27, using
(49−25, 49, 172)-Net in Base 27 — Constructive
(24, 49, 172)-net in base 27, using
- 19 times m-reduction [i] based on (24, 68, 172)-net in base 27, using
- base change [i] based on digital (7, 51, 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, 51, 172)-net over F81, using
(49−25, 49, 366)-Net over F27 — Digital
Digital (24, 49, 366)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2749, 366, F27, 2, 25) (dual of [(366, 2), 683, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2749, 732, F27, 25) (dual of [732, 683, 26]-code), using
- construction XX applied to C1 = C([727,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([727,23]) [i] based on
- linear OA(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [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(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(2745, 728, F27, 23) (dual of [728, 683, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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([727,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([727,23]) [i] based on
- OOA 2-folding [i] based on linear OA(2749, 732, F27, 25) (dual of [732, 683, 26]-code), using
(49−25, 49, 108098)-Net in Base 27 — Upper bound on s
There is no (24, 49, 108099)-net in base 27, because
- 1 times m-reduction [i] would yield (24, 48, 108099)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 507 563901 426176 225947 095647 568314 659311 665951 350720 588279 885187 125289 > 2748 [i]