Best Known (66, 66+21, s)-Nets in Base 5
(66, 66+21, 315)-Net over F5 — Constructive and digital
Digital (66, 87, 315)-net over F5, using
- net defined by OOA [i] based on linear OOA(587, 315, F5, 21, 21) (dual of [(315, 21), 6528, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(587, 3151, F5, 21) (dual of [3151, 3064, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(56, 26, F5, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(587, 3151, F5, 21) (dual of [3151, 3064, 22]-code), using
(66, 66+21, 2877)-Net over F5 — Digital
Digital (66, 87, 2877)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 2877, F5, 21) (dual of [2877, 2790, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 3148, F5, 21) (dual of [3148, 3061, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(54, 21, F5, 3) (dual of [21, 17, 4]-code or 21-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(587, 3148, F5, 21) (dual of [3148, 3061, 22]-code), using
(66, 66+21, 1161597)-Net in Base 5 — Upper bound on s
There is no (66, 87, 1161598)-net in base 5, because
- 1 times m-reduction [i] would yield (66, 86, 1161598)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 292473 381826 746010 092739 290819 448010 776353 439276 136196 844161 > 586 [i]