Best Known (77−19, 77, s)-Nets in Base 5
(77−19, 77, 348)-Net over F5 — Constructive and digital
Digital (58, 77, 348)-net over F5, using
- net defined by OOA [i] based on linear OOA(577, 348, F5, 19, 19) (dual of [(348, 19), 6535, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(577, 3133, F5, 19) (dual of [3133, 3056, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(577, 3133, F5, 19) (dual of [3133, 3056, 20]-code), using
(77−19, 77, 2380)-Net over F5 — Digital
Digital (58, 77, 2380)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 2380, F5, 19) (dual of [2380, 2303, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
(77−19, 77, 828135)-Net in Base 5 — Upper bound on s
There is no (58, 77, 828136)-net in base 5, because
- 1 times m-reduction [i] would yield (58, 76, 828136)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 132349 214054 147491 333087 114825 989240 239598 544305 154465 > 576 [i]