Best Known (194, 194+14, s)-Nets in Base 4
(194, 194+14, 4880872)-Net over F4 — Constructive and digital
Digital (194, 208, 4880872)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (41, 48, 87388)-net over F4, using
- net defined by OOA [i] based on linear OOA(448, 87388, F4, 7, 7) (dual of [(87388, 7), 611668, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(448, 262165, F4, 7) (dual of [262165, 262117, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(447, 262164, F4, 7) (dual of [262164, 262117, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(447, 262164, F4, 7) (dual of [262164, 262117, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(448, 262165, F4, 7) (dual of [262165, 262117, 8]-code), using
- net defined by OOA [i] based on linear OOA(448, 87388, F4, 7, 7) (dual of [(87388, 7), 611668, 8]-NRT-code), using
- digital (146, 160, 4793484)-net over F4, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- digital (41, 48, 87388)-net over F4, using
(194, 194+14, large)-Net over F4 — Digital
Digital (194, 208, large)-net over F4, using
- 41 times duplication [i] based on digital (193, 207, large)-net over F4, using
- t-expansion [i] based on digital (186, 207, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- t-expansion [i] based on digital (186, 207, large)-net over F4, using
(194, 194+14, large)-Net in Base 4 — Upper bound on s
There is no (194, 208, large)-net in base 4, because
- 12 times m-reduction [i] would yield (194, 196, large)-net in base 4, but