Best Known (60, 79, s)-Nets in Base 5
(60, 79, 349)-Net over F5 — Constructive and digital
Digital (60, 79, 349)-net over F5, using
- net defined by OOA [i] based on linear OOA(579, 349, F5, 19, 19) (dual of [(349, 19), 6552, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(579, 3142, F5, 19) (dual of [3142, 3063, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(579, 3143, F5, 19) (dual of [3143, 3064, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- 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(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(53, 18, F5, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(579, 3143, F5, 19) (dual of [3143, 3064, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(579, 3142, F5, 19) (dual of [3142, 3063, 20]-code), using
(60, 79, 2879)-Net over F5 — Digital
Digital (60, 79, 2879)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(579, 2879, F5, 19) (dual of [2879, 2800, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(579, 3143, F5, 19) (dual of [3143, 3064, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- 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(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(53, 18, F5, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(579, 3143, F5, 19) (dual of [3143, 3064, 20]-code), using
(60, 79, 1184211)-Net in Base 5 — Upper bound on s
There is no (60, 79, 1184212)-net in base 5, because
- 1 times m-reduction [i] would yield (60, 78, 1184212)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3 308739 959774 694034 535474 181056 853420 868369 223485 118545 > 578 [i]