Best Known (69−9, 69, s)-Nets in Base 4
(69−9, 69, 1048579)-Net over F4 — Constructive and digital
Digital (60, 69, 1048579)-net over F4, using
- 41 times duplication [i] based on digital (59, 68, 1048579)-net over F4, using
- net defined by OOA [i] based on linear OOA(468, 1048579, F4, 9, 9) (dual of [(1048579, 9), 9437143, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(468, 4194317, F4, 9) (dual of [4194317, 4194249, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(468, 4194317, F4, 9) (dual of [4194317, 4194249, 10]-code), using
- net defined by OOA [i] based on linear OOA(468, 1048579, F4, 9, 9) (dual of [(1048579, 9), 9437143, 10]-NRT-code), using
(69−9, 69, 2097159)-Net over F4 — Digital
Digital (60, 69, 2097159)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(469, 2097159, F4, 2, 9) (dual of [(2097159, 2), 4194249, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(469, 4194318, F4, 9) (dual of [4194318, 4194249, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(468, 4194317, F4, 9) (dual of [4194317, 4194249, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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(8) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(468, 4194317, F4, 9) (dual of [4194317, 4194249, 10]-code), using
- OOA 2-folding [i] based on linear OA(469, 4194318, F4, 9) (dual of [4194318, 4194249, 10]-code), using
(69−9, 69, large)-Net in Base 4 — Upper bound on s
There is no (60, 69, large)-net in base 4, because
- 7 times m-reduction [i] would yield (60, 62, large)-net in base 4, but