Best Known (61, 71, s)-Nets in Base 4
(61, 71, 209717)-Net over F4 — Constructive and digital
Digital (61, 71, 209717)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 209717, F4, 10, 10) (dual of [(209717, 10), 2097099, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
(61, 71, 524293)-Net over F4 — Digital
Digital (61, 71, 524293)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(471, 524293, F4, 2, 10) (dual of [(524293, 2), 1048515, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
(61, 71, large)-Net in Base 4 — Upper bound on s
There is no (61, 71, large)-net in base 4, because
- 8 times m-reduction [i] would yield (61, 63, large)-net in base 4, but