Projective Reed–Muller type codes on higher dimensional scrolls

PIR Codes with Short Block Length

“…for k 3. We remark that for k = 3 or k = 4 this inequality is always at least as good as the coding theoretic lower bound P(s, k) N(s, k) and it is indeed a strict improvement for larger values of s. 5 For systematic PIR codes, see the explanation below, the same bound was also proved in [29] and [27].…”

“…We call the ILP ( 9)-( 11), ( 13)-( 16) the the lower bound ILP for a given value of λ. We remark that the lower bound ILP for λ = 3 increases the coding theoretic lower bound by 1 in the cases (s, k) ∈ {(4, 3), (4, 4), (4, 12), (5, 10), (5,12), (6, 8), (6, 12)}, by 2 for (s, k) ∈ {(5, 8), (6, 14)}, and by 4 for (s, k) = (6, 16), cf. Table I.…”

“…6 However, this is not the case. An example is given by s = 92 and k = 5, where N(92, 5) = 106 7 but P(92, 5) 107 due to Inequality (5). Also for t > 6 there are such examples, however, they require rather large values of s. So, the situation should be as follows: If k > 4 and the dimension s is not too big with respect to k, then the coding theoretic lower bound is superior.…”

“…As hyperplane H we can choose the set of 15 non-zero vectors in F 5 2 that are perpendicular to the all-one vector (11111) :…”

“…I.e., the columns of a generator matrix G of C consist of all non-zero vectors of F 4 2 . Now consider any lengthened [17,5,8] code. The two new columns can be contained in at most two different recovery sets for e 5 .…”