Best Known (53, 60, s)-Nets in Base 4
(53, 60, 1398121)-Net over F4 — Constructive and digital
Digital (53, 60, 1398121)-net over F4, using
- net defined by OOA [i] based on linear OOA(460, 1398121, F4, 9, 7) (dual of [(1398121, 9), 12583029, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(460, 1398122, F4, 3, 7) (dual of [(1398122, 3), 4194306, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(44, 17, F4, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
- linear OOA(456, 1398105, F4, 3, 7) (dual of [(1398105, 3), 4194259, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(456, 4194315, F4, 7) (dual of [4194315, 4194259, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [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(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding [i] based on linear OA(456, 4194315, F4, 7) (dual of [4194315, 4194259, 8]-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(460, 1398122, F4, 3, 7) (dual of [(1398122, 3), 4194306, 8]-NRT-code), using
(53, 60, 4194334)-Net over F4 — Digital
Digital (53, 60, 4194334)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(460, 4194334, F4, 7) (dual of [4194334, 4194274, 8]-code), using
- (u, u+v)-construction [i] based on
- linear OA(43, 6, F4, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
- 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
- (u, u+v)-construction [i] based on
(53, 60, large)-Net in Base 4 — Upper bound on s
There is no (53, 60, large)-net in base 4, because
- 5 times m-reduction [i] would yield (53, 55, large)-net in base 4, but