Best Known (56−13, 56, s)-Nets in Base 5
(56−13, 56, 524)-Net over F5 — Constructive and digital
Digital (43, 56, 524)-net over F5, using
- net defined by OOA [i] based on linear OOA(556, 524, F5, 13, 13) (dual of [(524, 13), 6756, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(556, 3145, F5, 13) (dual of [3145, 3089, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 3146, F5, 13) (dual of [3146, 3090, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(556, 3146, F5, 13) (dual of [3146, 3090, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(556, 3145, F5, 13) (dual of [3145, 3089, 14]-code), using
(56−13, 56, 3207)-Net over F5 — Digital
Digital (43, 56, 3207)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 3207, F5, 13) (dual of [3207, 3151, 14]-code), using
- 72 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 41 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [i] based on
- 72 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 41 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
(56−13, 56, 1911552)-Net in Base 5 — Upper bound on s
There is no (43, 56, 1911553)-net in base 5, because
- 1 times m-reduction [i] would yield (43, 55, 1911553)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 277 556221 880952 137492 260562 988047 662345 > 555 [i]