Best Known (39−9, 39, s)-Nets in Base 5
(39−9, 39, 785)-Net over F5 — Constructive and digital
Digital (30, 39, 785)-net over F5, using
- net defined by OOA [i] based on linear OOA(539, 785, F5, 9, 9) (dual of [(785, 9), 7026, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(539, 3141, F5, 9) (dual of [3141, 3102, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(539, 3143, F5, 9) (dual of [3143, 3104, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(53, 18, F5, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(539, 3143, F5, 9) (dual of [3143, 3104, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(539, 3141, F5, 9) (dual of [3141, 3102, 10]-code), using
(39−9, 39, 3181)-Net over F5 — Digital
Digital (30, 39, 3181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(539, 3181, F5, 9) (dual of [3181, 3142, 10]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0) [i] based on linear OA(537, 3137, F5, 9) (dual of [3137, 3100, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- dual of repetition code with length 12 [i]
- linear OA(51, 12, F5, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 42 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0) [i] based on linear OA(537, 3137, F5, 9) (dual of [3137, 3100, 10]-code), using
(39−9, 39, 2416614)-Net in Base 5 — Upper bound on s
There is no (30, 39, 2416615)-net in base 5, because
- 1 times m-reduction [i] would yield (30, 38, 2416615)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 363 798067 754772 523564 457201 > 538 [i]