Best Known (230−15, 230, s)-Nets in Base 4
(230−15, 230, 6191593)-Net over F4 — Constructive and digital
Digital (215, 230, 6191593)-net over F4, using
- 41 times duplication [i] based on digital (214, 229, 6191593)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (50, 57, 1398109)-net over F4, using
- net defined by OOA [i] based on linear OOA(457, 1398109, F4, 7, 7) (dual of [(1398109, 7), 9786706, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(457, 4194328, F4, 7) (dual of [4194328, 4194271, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(456, 4194304, F4, 7) (dual of [4194304, 4194248, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(434, 4194304, F4, 5) (dual of [4194304, 4194270, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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
- OOA 3-folding and stacking with additional row [i] based on linear OA(457, 4194328, F4, 7) (dual of [4194328, 4194271, 8]-code), using
- net defined by OOA [i] based on linear OOA(457, 1398109, F4, 7, 7) (dual of [(1398109, 7), 9786706, 8]-NRT-code), using
- digital (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (50, 57, 1398109)-net over F4, using
- (u, u+v)-construction [i] based on
(230−15, 230, large)-Net over F4 — Digital
Digital (215, 230, large)-net over F4, using
- t-expansion [i] based on digital (213, 230, large)-net over F4, using
- 7 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 7 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
(230−15, 230, large)-Net in Base 4 — Upper bound on s
There is no (215, 230, large)-net in base 4, because
- 13 times m-reduction [i] would yield (215, 217, large)-net in base 4, but