Best Known (61, 61+19, s)-Nets in Base 5
(61, 61+19, 349)-Net over F5 — Constructive and digital
Digital (61, 80, 349)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(579, 349, F5, 19, 19) (dual of [(349, 19), 6552, 20]-NRT-code), using
(61, 61+19, 3154)-Net over F5 — Digital
Digital (61, 80, 3154)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 3154, F5, 19) (dual of [3154, 3074, 20]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 12 times 0) [i] based on linear OA(576, 3130, F5, 19) (dual of [3130, 3054, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [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(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(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(18) ⊂ Ce(17) [i] based on
- 20 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 12 times 0) [i] based on linear OA(576, 3130, F5, 19) (dual of [3130, 3054, 20]-code), using
(61, 61+19, 1416096)-Net in Base 5 — Upper bound on s
There is no (61, 80, 1416097)-net in base 5, because
- 1 times m-reduction [i] would yield (61, 79, 1416097)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 16 543643 565431 180152 172535 715874 013332 696825 897931 164645 > 579 [i]