Best Known (61, 61+14, s)-Nets in Base 5
(61, 61+14, 2243)-Net over F5 — Constructive and digital
Digital (61, 75, 2243)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (53, 67, 2233)-net over F5, using
- net defined by OOA [i] based on linear OOA(567, 2233, F5, 14, 14) (dual of [(2233, 14), 31195, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(567, 15631, F5, 14) (dual of [15631, 15564, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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
- OA 7-folding and stacking [i] based on linear OA(567, 15631, F5, 14) (dual of [15631, 15564, 15]-code), using
- net defined by OOA [i] based on linear OOA(567, 2233, F5, 14, 14) (dual of [(2233, 14), 31195, 15]-NRT-code), using
- digital (1, 8, 10)-net over F5, using
(61, 61+14, 15659)-Net over F5 — Digital
Digital (61, 75, 15659)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(575, 15659, F5, 14) (dual of [15659, 15584, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(573, 15655, F5, 14) (dual of [15655, 15582, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(573, 15657, F5, 13) (dual of [15657, 15584, 14]-code), using Gilbert–Varšamov bound and bm = 573 > Vbs−1(k−1) = 7 564926 142226 848504 187654 465481 999243 875916 317025 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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
- linear OA(573, 15655, F5, 14) (dual of [15655, 15582, 15]-code), using
- construction X with Varšamov bound [i] based on
(61, 61+14, large)-Net in Base 5 — Upper bound on s
There is no (61, 75, large)-net in base 5, because
- 12 times m-reduction [i] would yield (61, 63, large)-net in base 5, but