Best Known (48, 48+41, s)-Nets in Base 27
(48, 48+41, 210)-Net over F27 — Constructive and digital
Digital (48, 89, 210)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 17, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (4, 24, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (7, 48, 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 (4, 17, 64)-net over F27, using
(48, 48+41, 370)-Net in Base 27 — Constructive
(48, 89, 370)-net in base 27, using
- t-expansion [i] based on (43, 89, 370)-net in base 27, using
- 19 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
- 19 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(48, 48+41, 949)-Net over F27 — Digital
Digital (48, 89, 949)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2789, 949, F27, 41) (dual of [949, 860, 42]-code), using
- 206 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 21 times 0, 1, 38 times 0, 1, 55 times 0, 1, 69 times 0) [i] based on linear OA(2778, 732, F27, 41) (dual of [732, 654, 42]-code), using
- construction XX applied to C1 = C([727,38]), C2 = C([0,39]), C3 = C1 + C2 = C([0,38]), and C∩ = C1 ∩ C2 = C([727,39]) [i] based on
- 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(2776, 728, F27, 40) (dual of [728, 652, 41]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,39], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2778, 728, F27, 41) (dual of [728, 650, 42]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,39}, and designed minimum distance d ≥ |I|+1 = 42 [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(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([727,38]), C2 = C([0,39]), C3 = C1 + C2 = C([0,38]), and C∩ = C1 ∩ C2 = C([727,39]) [i] based on
- 206 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 21 times 0, 1, 38 times 0, 1, 55 times 0, 1, 69 times 0) [i] based on linear OA(2778, 732, F27, 41) (dual of [732, 654, 42]-code), using
(48, 48+41, 634346)-Net in Base 27 — Upper bound on s
There is no (48, 89, 634347)-net in base 27, because
- 1 times m-reduction [i] would yield (48, 88, 634347)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 912054 436116 858971 810253 729774 168592 047094 094942 913705 676360 348825 586380 600845 719990 493323 204761 743978 832839 838805 417280 752601 > 2788 [i]