Best Known (68−16, 68, s)-Nets in Base 5
(68−16, 68, 394)-Net over F5 — Constructive and digital
Digital (52, 68, 394)-net over F5, using
- net defined by OOA [i] based on linear OOA(568, 394, F5, 16, 16) (dual of [(394, 16), 6236, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(568, 3152, F5, 16) (dual of [3152, 3084, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(567, 3151, F5, 16) (dual of [3151, 3084, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 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(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(15) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(567, 3151, F5, 16) (dual of [3151, 3084, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(568, 3152, F5, 16) (dual of [3152, 3084, 17]-code), using
(68−16, 68, 3201)-Net over F5 — Digital
Digital (52, 68, 3201)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(568, 3201, F5, 16) (dual of [3201, 3133, 17]-code), using
- 69 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on 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]
- 69 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
(68−16, 68, 822000)-Net in Base 5 — Upper bound on s
There is no (52, 68, 822001)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 338814 057431 007445 604705 266965 354132 186699 526625 > 568 [i]