Best Known (80, 80+26, s)-Nets in Base 5
(80, 80+26, 400)-Net over F5 — Constructive and digital
Digital (80, 106, 400)-net over F5, using
- 4 times m-reduction [i] based on digital (80, 110, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 55, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 55, 200)-net over F25, using
(80, 80+26, 2784)-Net over F5 — Digital
Digital (80, 106, 2784)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5106, 2784, F5, 26) (dual of [2784, 2678, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, 3146, F5, 26) (dual of [3146, 3040, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5106, 3146, F5, 26) (dual of [3146, 3040, 27]-code), using
(80, 80+26, 708980)-Net in Base 5 — Upper bound on s
There is no (80, 106, 708981)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 123 260233 831215 366616 877920 111475 580585 082811 597827 883973 374781 481085 722725 > 5106 [i]