Best Known (39, 39+12, s)-Nets in Base 5
(39, 39+12, 524)-Net over F5 — Constructive and digital
Digital (39, 51, 524)-net over F5, using
- 51 times duplication [i] based on digital (38, 50, 524)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 524, F5, 12, 12) (dual of [(524, 12), 6238, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
- net defined by OOA [i] based on linear OOA(550, 524, F5, 12, 12) (dual of [(524, 12), 6238, 13]-NRT-code), using
(39, 39+12, 3181)-Net over F5 — Digital
Digital (39, 51, 3181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(551, 3181, F5, 12) (dual of [3181, 3130, 13]-code), using
- 46 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 26 times 0) [i] based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(50, 5, F5, 0) (dual of [5, 5, 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 X applied to Ce(11) ⊂ Ce(10) [i] based on
- 46 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 26 times 0) [i] based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
(39, 39+12, 653739)-Net in Base 5 — Upper bound on s
There is no (39, 51, 653740)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 444092 815636 895735 779866 346044 263233 > 551 [i]