Best Known (50, 50+46, s)-Nets in Base 27
(50, 50+46, 196)-Net over F27 — Constructive and digital
Digital (50, 96, 196)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 32, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (18, 64, 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 (9, 32, 88)-net over F27, using
(50, 50+46, 370)-Net in Base 27 — Constructive
(50, 96, 370)-net in base 27, using
- t-expansion [i] based on (43, 96, 370)-net in base 27, using
- 12 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
- 12 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(50, 50+46, 797)-Net over F27 — Digital
Digital (50, 96, 797)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2796, 797, F27, 46) (dual of [797, 701, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(2796, 800, F27, 46) (dual of [800, 704, 47]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 31 times 0) [i] based on linear OA(2788, 732, F27, 46) (dual of [732, 644, 47]-code), using
- construction XX applied to C1 = C([727,43]), C2 = C([0,44]), C3 = C1 + C2 = C([0,43]), and C∩ = C1 ∩ C2 = C([727,44]) [i] based on
- linear OA(2786, 728, F27, 45) (dual of [728, 642, 46]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,43}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(2786, 728, F27, 45) (dual of [728, 642, 46]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,44], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(2788, 728, F27, 46) (dual of [728, 640, 47]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,44}, and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2784, 728, F27, 44) (dual of [728, 644, 45]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,43], and designed minimum distance d ≥ |I|+1 = 45 [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([727,43]), C2 = C([0,44]), C3 = C1 + C2 = C([0,43]), and C∩ = C1 ∩ C2 = C([727,44]) [i] based on
- 60 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 31 times 0) [i] based on linear OA(2788, 732, F27, 46) (dual of [732, 644, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(2796, 800, F27, 46) (dual of [800, 704, 47]-code), using
(50, 50+46, 341865)-Net in Base 27 — Upper bound on s
There is no (50, 96, 341866)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 257588 077605 619430 825466 261920 060852 880109 873803 356031 497395 768970 786064 126611 378871 813491 390547 314577 870323 230113 464291 944997 000201 144601 > 2796 [i]