Best Known (154−14, 154, s)-Nets in Base 4
(154−14, 154, 1199741)-Net over F4 — Constructive and digital
Digital (140, 154, 1199741)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 33, 1370)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 1370, F4, 7, 7) (dual of [(1370, 7), 9557, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(433, 4111, F4, 7) (dual of [4111, 4078, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(432, 4110, F4, 7) (dual of [4110, 4078, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(413, 14, F4, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,4)), using
- dual of repetition code with length 14 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 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(432, 4110, F4, 7) (dual of [4110, 4078, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(433, 4111, F4, 7) (dual of [4111, 4078, 8]-code), using
- net defined by OOA [i] based on linear OOA(433, 1370, F4, 7, 7) (dual of [(1370, 7), 9557, 8]-NRT-code), using
- digital (107, 121, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 1198371, F4, 14, 14) (dual of [(1198371, 14), 16777073, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4121, 8388597, F4, 14) (dual of [8388597, 8388476, 15]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4121, 8388597, F4, 14) (dual of [8388597, 8388476, 15]-code), using
- net defined by OOA [i] based on linear OOA(4121, 1198371, F4, 14, 14) (dual of [(1198371, 14), 16777073, 15]-NRT-code), using
- digital (26, 33, 1370)-net over F4, using
(154−14, 154, large)-Net over F4 — Digital
Digital (140, 154, large)-net over F4, using
- t-expansion [i] based on digital (139, 154, large)-net over F4, using
- 1 times m-reduction [i] based on digital (139, 155, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 1 times m-reduction [i] based on digital (139, 155, large)-net over F4, using
(154−14, 154, large)-Net in Base 4 — Upper bound on s
There is no (140, 154, large)-net in base 4, because
- 12 times m-reduction [i] would yield (140, 142, large)-net in base 4, but