Best Known (22, 40, s)-Nets in Base 27
(22, 40, 158)-Net over F27 — Constructive and digital
Digital (22, 40, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 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 (7, 25, 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 (6, 15, 76)-net over F27, using
(22, 40, 216)-Net in Base 27 — Constructive
(22, 40, 216)-net in base 27, using
- base change [i] based on digital (12, 30, 216)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (2, 20, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (1, 10, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(22, 40, 773)-Net over F27 — Digital
Digital (22, 40, 773)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2740, 773, F27, 18) (dual of [773, 733, 19]-code), using
- 36 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 24 times 0) [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
- 36 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 24 times 0) [i] based on linear OA(2735, 732, F27, 18) (dual of [732, 697, 19]-code), using
(22, 40, 366766)-Net in Base 27 — Upper bound on s
There is no (22, 40, 366767)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1797 031509 107956 572800 819911 285760 560017 570523 980380 364535 > 2740 [i]