Best Known (51, 65, s)-Nets in Base 5
(51, 65, 459)-Net over F5 — Constructive and digital
Digital (51, 65, 459)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (42, 56, 447)-net over F5, using
- net defined by OOA [i] based on linear OOA(556, 447, F5, 14, 14) (dual of [(447, 14), 6202, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(556, 3129, F5, 14) (dual of [3129, 3073, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(556, 3129, F5, 14) (dual of [3129, 3073, 15]-code), using
- net defined by OOA [i] based on linear OOA(556, 447, F5, 14, 14) (dual of [(447, 14), 6202, 15]-NRT-code), using
- digital (2, 9, 12)-net over F5, using
(51, 65, 4457)-Net over F5 — Digital
Digital (51, 65, 4457)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(565, 4457, F5, 14) (dual of [4457, 4392, 15]-code), using
- 1318 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0, 1, 236 times 0, 1, 359 times 0, 1, 472 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- 1318 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0, 1, 236 times 0, 1, 359 times 0, 1, 472 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
(51, 65, 2613928)-Net in Base 5 — Upper bound on s
There is no (51, 65, 2613929)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 2710 512636 736065 381094 611192 483804 650472 026445 > 565 [i]