Best Known (259−36, 259, s)-Nets in Base 4
(259−36, 259, 14567)-Net over F4 — Constructive and digital
Digital (223, 259, 14567)-net over F4, using
- 1 times m-reduction [i] based on digital (223, 260, 14567)-net over F4, using
- net defined by OOA [i] based on linear OOA(4260, 14567, F4, 37, 37) (dual of [(14567, 37), 538719, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4260, 262207, F4, 37) (dual of [262207, 261947, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4260, 262208, F4, 37) (dual of [262208, 261948, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4244, 262144, F4, 37) (dual of [262144, 261900, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4260, 262208, F4, 37) (dual of [262208, 261948, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4260, 262207, F4, 37) (dual of [262207, 261947, 38]-code), using
- net defined by OOA [i] based on linear OOA(4260, 14567, F4, 37, 37) (dual of [(14567, 37), 538719, 38]-NRT-code), using
(259−36, 259, 167054)-Net over F4 — Digital
Digital (223, 259, 167054)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4259, 167054, F4, 36) (dual of [167054, 166795, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 262213, F4, 36) (dual of [262213, 261954, 37]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4257, 262211, F4, 36) (dual of [262211, 261954, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4244, 262144, F4, 37) (dual of [262144, 261900, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(413, 67, F4, 6) (dual of [67, 54, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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(5) ⊂ Ce(4) [i] based on
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4257, 262211, F4, 36) (dual of [262211, 261954, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 262213, F4, 36) (dual of [262213, 261954, 37]-code), using
(259−36, 259, large)-Net in Base 4 — Upper bound on s
There is no (223, 259, large)-net in base 4, because
- 34 times m-reduction [i] would yield (223, 225, large)-net in base 4, but