Best Known (92−44, 92, s)-Nets in Base 27
(92−44, 92, 192)-Net over F27 — Constructive and digital
Digital (48, 92, 192)-net over F27, using
- 8 times m-reduction [i] based on digital (48, 100, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 37, 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, 63, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 37, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(92−44, 92, 370)-Net in Base 27 — Constructive
(48, 92, 370)-net in base 27, using
- t-expansion [i] based on (43, 92, 370)-net in base 27, using
- 16 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
- 16 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(92−44, 92, 782)-Net over F27 — Digital
Digital (48, 92, 782)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2792, 782, F27, 44) (dual of [782, 690, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(2792, 789, F27, 44) (dual of [789, 697, 45]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 4 times 0, 1, 13 times 0, 1, 26 times 0) [i] based on linear OA(2784, 732, F27, 44) (dual of [732, 648, 45]-code), using
- construction XX applied to C1 = C([727,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([727,42]) [i] based on
- linear OA(2782, 728, F27, 43) (dual of [728, 646, 44]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,41}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2782, 728, F27, 43) (dual of [728, 646, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2784, 728, F27, 44) (dual of [728, 644, 45]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,42}, and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2780, 728, F27, 42) (dual of [728, 648, 43]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,41], and designed minimum distance d ≥ |I|+1 = 43 [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([727,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([727,42]) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 4 times 0, 1, 13 times 0, 1, 26 times 0) [i] based on linear OA(2784, 732, F27, 44) (dual of [732, 648, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(2792, 789, F27, 44) (dual of [789, 697, 45]-code), using
(92−44, 92, 336952)-Net in Base 27 — Upper bound on s
There is no (48, 92, 336953)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 484723 397666 328070 243440 034463 148650 198737 524935 815605 966181 555435 525504 405599 784274 117019 233782 837062 576874 419748 547288 996261 869113 > 2792 [i]