Best Known (53, 53+48, s)-Nets in Base 27
(53, 53+48, 204)-Net over F27 — Constructive and digital
Digital (53, 101, 204)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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 (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 (11, 35, 96)-net over F27, using
(53, 53+48, 370)-Net in Base 27 — Constructive
(53, 101, 370)-net in base 27, using
- t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 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
- 7 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(53, 53+48, 866)-Net over F27 — Digital
Digital (53, 101, 866)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27101, 866, F27, 48) (dual of [866, 765, 49]-code), using
- 125 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 36 times 0, 1, 50 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
- 125 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 36 times 0, 1, 50 times 0) [i] based on linear OA(2792, 732, F27, 48) (dual of [732, 640, 49]-code), using
(53, 53+48, 398136)-Net in Base 27 — Upper bound on s
There is no (53, 101, 398137)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 3 696245 911891 708988 730364 057897 478598 194272 500354 384094 888419 358320 062384 028811 251022 123116 424059 310772 171701 022093 155094 735456 180845 079778 624417 > 27101 [i]