Best Known (52, 52+48, s)-Nets in Base 27
(52, 52+48, 202)-Net over F27 — Constructive and digital
Digital (52, 100, 202)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 34, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (18, 66, 108)-net over F27, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- F3 from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F27 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- digital (10, 34, 94)-net over F27, using
(52, 52+48, 370)-Net in Base 27 — Constructive
(52, 100, 370)-net in base 27, using
- t-expansion [i] based on (43, 100, 370)-net in base 27, using
- 8 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
- 8 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(52, 52+48, 812)-Net over F27 — Digital
Digital (52, 100, 812)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27100, 812, F27, 48) (dual of [812, 712, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(27100, 814, F27, 48) (dual of [814, 714, 49]-code), using
- 74 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 36 times 0) [i] based on linear OA(2792, 732, F27, 48) (dual of [732, 640, 49]-code), using
- construction XX applied to C1 = C([727,45]), C2 = C([0,46]), C3 = C1 + C2 = C([0,45]), and C∩ = C1 ∩ C2 = C([727,46]) [i] based on
- linear OA(2790, 728, F27, 47) (dual of [728, 638, 48]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,45}, and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(2790, 728, F27, 47) (dual of [728, 638, 48]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,46], and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(2792, 728, F27, 48) (dual of [728, 636, 49]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,46}, and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(2788, 728, F27, 46) (dual of [728, 640, 47]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,45], and designed minimum distance d ≥ |I|+1 = 47 [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([727,45]), C2 = C([0,46]), C3 = C1 + C2 = C([0,45]), and C∩ = C1 ∩ C2 = C([727,46]) [i] based on
- 74 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 36 times 0) [i] based on linear OA(2792, 732, F27, 48) (dual of [732, 640, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(27100, 814, F27, 48) (dual of [814, 714, 49]-code), using
(52, 52+48, 347047)-Net in Base 27 — Upper bound on s
There is no (52, 100, 347048)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 136891 626243 824842 797960 910326 496377 014107 807692 257043 878891 354902 477008 238884 139669 171678 054489 491590 605303 681404 513923 041941 786803 123651 525889 > 27100 [i]