Best Known (33, 33+9, s)-Nets in Base 5
(33, 33+9, 976)-Net over F5 — Constructive and digital
Digital (33, 42, 976)-net over F5, using
- net defined by OOA [i] based on linear OOA(542, 976, F5, 9, 9) (dual of [(976, 9), 8742, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(542, 3905, F5, 9) (dual of [3905, 3863, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(542, 3906, F5, 9) (dual of [3906, 3864, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(542, 3905, F5, 9) (dual of [3905, 3863, 10]-code), using
(33, 33+9, 4437)-Net over F5 — Digital
Digital (33, 42, 4437)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(542, 4437, F5, 9) (dual of [4437, 4395, 10]-code), using
- 1295 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0, 1, 144 times 0, 1, 402 times 0, 1, 704 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
- 1295 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0, 1, 144 times 0, 1, 402 times 0, 1, 704 times 0) [i] based on linear OA(537, 3137, F5, 9) (dual of [3137, 3100, 10]-code), using
(33, 33+9, 8080444)-Net in Base 5 — Upper bound on s
There is no (33, 42, 8080445)-net in base 5, because
- 1 times m-reduction [i] would yield (33, 41, 8080445)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 45474 755772 445667 093761 775601 > 541 [i]