Best Known (50, 97, s)-Nets in Base 27
(50, 97, 196)-Net over F27 — Constructive and digital
Digital (50, 97, 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, 65, 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, 97, 370)-Net in Base 27 — Constructive
(50, 97, 370)-net in base 27, using
- t-expansion [i] based on (43, 97, 370)-net in base 27, using
- 11 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
- 11 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(50, 97, 746)-Net over F27 — Digital
Digital (50, 97, 746)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2797, 746, F27, 47) (dual of [746, 649, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(2797, 752, F27, 47) (dual of [752, 655, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(38) [i] based on
- linear OA(2790, 729, F27, 47) (dual of [729, 639, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2774, 729, F27, 39) (dual of [729, 655, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(277, 23, F27, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,27)), using
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- Reed–Solomon code RS(20,27) [i]
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- construction X applied to Ce(46) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(2797, 752, F27, 47) (dual of [752, 655, 48]-code), using
(50, 97, 341865)-Net in Base 27 — Upper bound on s
There is no (50, 97, 341866)-net in base 27, because
- 1 times m-reduction [i] would yield (50, 96, 341866)-net in base 27, but
- 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]