Best Known (83, 83+14, s)-Nets in Base 4
(83, 83+14, 37453)-Net over F4 — Constructive and digital
Digital (83, 97, 37453)-net over F4, using
- 41 times duplication [i] based on digital (82, 96, 37453)-net over F4, using
- net defined by OOA [i] based on linear OOA(496, 37453, F4, 14, 14) (dual of [(37453, 14), 524246, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(496, 262171, F4, 14) (dual of [262171, 262075, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 262176, F4, 14) (dual of [262176, 262080, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(496, 262176, F4, 14) (dual of [262176, 262080, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(496, 262171, F4, 14) (dual of [262171, 262075, 15]-code), using
- net defined by OOA [i] based on linear OOA(496, 37453, F4, 14, 14) (dual of [(37453, 14), 524246, 15]-NRT-code), using
(83, 83+14, 131088)-Net over F4 — Digital
Digital (83, 97, 131088)-net over F4, using
- 41 times duplication [i] based on digital (82, 96, 131088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(496, 131088, F4, 2, 14) (dual of [(131088, 2), 262080, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(496, 262176, F4, 14) (dual of [262176, 262080, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(496, 262176, F4, 14) (dual of [262176, 262080, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(496, 131088, F4, 2, 14) (dual of [(131088, 2), 262080, 15]-NRT-code), using
(83, 83+14, large)-Net in Base 4 — Upper bound on s
There is no (83, 97, large)-net in base 4, because
- 12 times m-reduction [i] would yield (83, 85, large)-net in base 4, but