Best Known (77−11, 77, s)-Nets in Base 5
(77−11, 77, 390630)-Net over F5 — Constructive and digital
Digital (66, 77, 390630)-net over F5, using
- net defined by OOA [i] based on linear OOA(577, 390630, F5, 11, 11) (dual of [(390630, 11), 4296853, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(577, 1953151, F5, 11) (dual of [1953151, 1953074, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(577, 1953151, F5, 11) (dual of [1953151, 1953074, 12]-code), using
(77−11, 77, 976575)-Net over F5 — Digital
Digital (66, 77, 976575)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(577, 976575, F5, 2, 11) (dual of [(976575, 2), 1953073, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(577, 1953150, F5, 11) (dual of [1953150, 1953073, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 1953151, F5, 11) (dual of [1953151, 1953074, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 1953151, F5, 11) (dual of [1953151, 1953074, 12]-code), using
- OOA 2-folding [i] based on linear OA(577, 1953150, F5, 11) (dual of [1953150, 1953073, 12]-code), using
(77−11, 77, large)-Net in Base 5 — Upper bound on s
There is no (66, 77, large)-net in base 5, because
- 9 times m-reduction [i] would yield (66, 68, large)-net in base 5, but