Best Known (48, 48+14, s)-Nets in Base 5
(48, 48+14, 450)-Net over F5 — Constructive and digital
Digital (48, 62, 450)-net over F5, using
- net defined by OOA [i] based on linear OOA(562, 450, F5, 14, 14) (dual of [(450, 14), 6238, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(562, 3150, F5, 14) (dual of [3150, 3088, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(562, 3151, F5, 14) (dual of [3151, 3089, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(562, 3151, F5, 14) (dual of [3151, 3089, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(562, 3150, F5, 14) (dual of [3150, 3088, 15]-code), using
(48, 48+14, 3384)-Net over F5 — Digital
Digital (48, 62, 3384)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(562, 3384, F5, 14) (dual of [3384, 3322, 15]-code), using
- 248 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) [i] based on
- 248 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
(48, 48+14, 1311397)-Net in Base 5 — Upper bound on s
There is no (48, 62, 1311398)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 21 684131 540447 902745 479174 487611 942711 062201 > 562 [i]