Best Known (58, 58+18, s)-Nets in Base 5
(58, 58+18, 349)-Net over F5 — Constructive and digital
Digital (58, 76, 349)-net over F5, using
- 51 times duplication [i] based on digital (57, 75, 349)-net over F5, using
- net defined by OOA [i] based on linear OOA(575, 349, F5, 18, 18) (dual of [(349, 18), 6207, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(575, 3141, F5, 18) (dual of [3141, 3066, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(575, 3141, F5, 18) (dual of [3141, 3066, 19]-code), using
- net defined by OOA [i] based on linear OOA(575, 349, F5, 18, 18) (dual of [(349, 18), 6207, 19]-NRT-code), using
(58, 58+18, 3162)-Net over F5 — Digital
Digital (58, 76, 3162)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(576, 3162, F5, 18) (dual of [3162, 3086, 19]-code), using
- 22 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- 22 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 13 times 0) [i] based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
(58, 58+18, 828135)-Net in Base 5 — Upper bound on s
There is no (58, 76, 828136)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 132349 214054 147491 333087 114825 989240 239598 544305 154465 > 576 [i]