Best Known (85−40, 85, s)-Nets in Base 27
(85−40, 85, 192)-Net over F27 — Constructive and digital
Digital (45, 85, 192)-net over F27, using
- 6 times m-reduction [i] based on digital (45, 91, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (11, 57, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 34, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(85−40, 85, 370)-Net in Base 27 — Constructive
(45, 85, 370)-net in base 27, using
- t-expansion [i] based on (43, 85, 370)-net in base 27, using
- 23 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- 23 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(85−40, 85, 817)-Net over F27 — Digital
Digital (45, 85, 817)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2785, 817, F27, 40) (dual of [817, 732, 41]-code), using
- 76 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 36 times 0) [i] based on linear OA(2776, 732, F27, 40) (dual of [732, 656, 41]-code), using
- construction XX applied to C1 = C([727,37]), C2 = C([0,38]), C3 = C1 + C2 = C([0,37]), and C∩ = C1 ∩ C2 = C([727,38]) [i] based on
- linear OA(2774, 728, F27, 39) (dual of [728, 654, 40]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,37}, and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2774, 728, F27, 39) (dual of [728, 654, 40]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,38], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2776, 728, F27, 40) (dual of [728, 652, 41]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,38}, and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2772, 728, F27, 38) (dual of [728, 656, 39]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,37], and designed minimum distance d ≥ |I|+1 = 39 [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([727,37]), C2 = C([0,38]), C3 = C1 + C2 = C([0,37]), and C∩ = C1 ∩ C2 = C([727,38]) [i] based on
- 76 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 36 times 0) [i] based on linear OA(2776, 732, F27, 40) (dual of [732, 656, 41]-code), using
(85−40, 85, 386916)-Net in Base 27 — Upper bound on s
There is no (45, 85, 386917)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 46 337127 429188 455800 511566 992994 474540 845296 081534 128361 046655 524019 303091 991893 416781 181251 935359 387150 421964 510538 812977 > 2785 [i]