Best Known (12−3, 12, s)-Nets in Base 4
(12−3, 12, 34566)-Net over F4 — Constructive and digital
Digital (9, 12, 34566)-net over F4, using
- net defined by OOA [i] based on linear OOA(412, 34566, F4, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(412, 34566, F4, 3) (dual of [34566, 34554, 4]-code or 34566-cap in PG(11,4)), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
(12−3, 12, 430674)-Net over F4 — Upper bound on s (digital)
There is no digital (9, 12, 430675)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(412, 430675, F4, 3) (dual of [430675, 430663, 4]-code or 430675-cap in PG(11,4)), but
- removing affine subspaces [i] would yield
- linear OA(47, 608, F4, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
- 1 times Hill recurrence [i] would yield linear OA(46, 154, F4, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
- construction Y1 [i] would yield
- linear OA(45, 42, F4, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
- linear OA(4148, 154, F4, 112) (dual of [154, 6, 113]-code), but
- discarding factors / shortening the dual code would yield linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- residual code [i] would yield linear OA(48, 11, F4, 7) (dual of [11, 3, 8]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- discarding factors / shortening the dual code would yield linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- construction Y1 [i] would yield
- 1 times Hill recurrence [i] would yield linear OA(46, 154, F4, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
- 1752-cap in AG(7,4), but
- 3 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4), but
- 6329-cap in AG(8,4), but
- 4 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- 23085-cap in AG(9,4), but
- 5 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- 84865-cap in AG(10,4), but
- 6 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- 314041-cap in AG(11,4), but
- 7 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- linear OA(47, 608, F4, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
- removing affine subspaces [i] would yield
(12−3, 12, 1398100)-Net in Base 4 — Upper bound on s
There is no (9, 12, 1398101)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(412, 1398101, S4, 3), but