Best Known (39, 77, s)-Nets in Base 25
(39, 77, 252)-Net over F25 — Constructive and digital
Digital (39, 77, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 29, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (10, 48, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 29, 126)-net over F25, using
(39, 77, 516)-Net over F25 — Digital
Digital (39, 77, 516)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2577, 516, F25, 38) (dual of [516, 439, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2577, 642, F25, 38) (dual of [642, 565, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- linear OA(2572, 625, F25, 38) (dual of [625, 553, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2560, 625, F25, 32) (dual of [625, 565, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(255, 17, F25, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(2577, 642, F25, 38) (dual of [642, 565, 39]-code), using
(39, 77, 152866)-Net in Base 25 — Upper bound on s
There is no (39, 77, 152867)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 437933 936364 559071 905864 863007 158596 688706 792702 044282 038302 356896 959414 513108 459642 068328 682065 387448 580825 > 2577 [i]