Best Known (66, 88, s)-Nets in Base 5
(66, 88, 306)-Net over F5 — Constructive and digital
Digital (66, 88, 306)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (13, 24, 54)-net over F5, using
- trace code for nets [i] based on digital (1, 12, 27)-net over F25, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
- trace code for nets [i] based on digital (1, 12, 27)-net over F25, using
- digital (42, 64, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 32, 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
- trace code for nets [i] based on digital (10, 32, 126)-net over F25, using
- digital (13, 24, 54)-net over F5, using
(66, 88, 2265)-Net over F5 — Digital
Digital (66, 88, 2265)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(588, 2265, F5, 22) (dual of [2265, 2177, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(588, 3133, F5, 22) (dual of [3133, 3045, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(588, 3133, F5, 22) (dual of [3133, 3045, 23]-code), using
(66, 88, 479410)-Net in Base 5 — Upper bound on s
There is no (66, 88, 479411)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 32 312410 208429 411980 880729 585124 632308 071802 469224 565788 572005 > 588 [i]