Best Known (123, 123+12, s)-Nets in Base 4
(123, 123+12, 4194301)-Net over F4 — Constructive and digital
Digital (123, 135, 4194301)-net over F4, using
- net defined by OOA [i] based on linear OOA(4135, 4194301, F4, 15, 12) (dual of [(4194301, 15), 62914380, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(4135, large, F4, 3, 12), using
- trace code [i] based on linear OOA(6445, 2796201, F64, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- trace code [i] based on linear OOA(6445, 2796201, F64, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(4135, large, F4, 3, 12), using
(123, 123+12, large)-Net over F4 — Digital
Digital (123, 135, large)-net over F4, using
- t-expansion [i] based on digital (121, 135, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
(123, 123+12, large)-Net in Base 4 — Upper bound on s
There is no (123, 135, large)-net in base 4, because
- 10 times m-reduction [i] would yield (123, 125, large)-net in base 4, but