Best Known (99−22, 99, s)-Nets in Base 5
(99−22, 99, 400)-Net over F5 — Constructive and digital
Digital (77, 99, 400)-net over F5, using
- 5 times m-reduction [i] based on digital (77, 104, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 52, 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, 52, 200)-net over F25, using
(99−22, 99, 4300)-Net over F5 — Digital
Digital (77, 99, 4300)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 4300, F5, 22) (dual of [4300, 4201, 23]-code), using
- 1157 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0, 1, 258 times 0, 1, 300 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [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(50, 5, F5, 0) (dual of [5, 5, 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
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 1157 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0, 1, 258 times 0, 1, 300 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
(99−22, 99, 2397081)-Net in Base 5 — Upper bound on s
There is no (77, 99, 2397082)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1577 725445 991552 976974 357526 086497 044982 282483 784390 681858 919643 332105 > 599 [i]