Best Known (244−35, 244, s)-Nets in Base 4
(244−35, 244, 15422)-Net over F4 — Constructive and digital
Digital (209, 244, 15422)-net over F4, using
- 44 times duplication [i] based on digital (205, 240, 15422)-net over F4, using
- net defined by OOA [i] based on linear OOA(4240, 15422, F4, 35, 35) (dual of [(15422, 35), 539530, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4240, 262175, F4, 35) (dual of [262175, 261935, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 262176, F4, 35) (dual of [262176, 261936, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4240, 262176, F4, 35) (dual of [262176, 261936, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4240, 262175, F4, 35) (dual of [262175, 261935, 36]-code), using
- net defined by OOA [i] based on linear OOA(4240, 15422, F4, 35, 35) (dual of [(15422, 35), 539530, 36]-NRT-code), using
(244−35, 244, 131097)-Net over F4 — Digital
Digital (209, 244, 131097)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4244, 131097, F4, 2, 35) (dual of [(131097, 2), 261950, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4244, 262194, F4, 35) (dual of [262194, 261950, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4244, 262195, F4, 35) (dual of [262195, 261951, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4244, 262195, F4, 35) (dual of [262195, 261951, 36]-code), using
- OOA 2-folding [i] based on linear OA(4244, 262194, F4, 35) (dual of [262194, 261950, 36]-code), using
(244−35, 244, large)-Net in Base 4 — Upper bound on s
There is no (209, 244, large)-net in base 4, because
- 33 times m-reduction [i] would yield (209, 211, large)-net in base 4, but