Best Known (260−37, 260, s)-Nets in Base 4
(260−37, 260, 14567)-Net over F4 — Constructive and digital
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
(260−37, 260, 132219)-Net over F4 — Digital
Digital (223, 260, 132219)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4260, 132219, F4, 37) (dual of [132219, 131959, 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
(260−37, 260, large)-Net in Base 4 — Upper bound on s
There is no (223, 260, large)-net in base 4, because
- 35 times m-reduction [i] would yield (223, 225, large)-net in base 4, but