On Hyperbolic Codes

On the Feng-Rao Bound for Generalized Hamming Weights

“…It is also known that (12) gives the actual values of the tth generalized Hamming weights of the Hermitian codes (see [2]). For the case of hyperbolic codes (improved q-ary Reed-Muller codes) (13) gives exactly the same estimates as was found in [5]. We note that the result concerning the condition for tth rank MDS from the previous section is easily translated into the setting of the present section.…”

“…Clearly, this code has parameters [n, k, d] = [8,5,3]. We now show that for the particular choice of B our new bound (5) will give us at least d(C(3)) ≥ 2 whereas the Shibuya-Sakaniwa bound will only give d(C(3)) ≥ 1.…”

“…We postpone a discussion of these connections to the next section. Obviously the result concerning the code C ⊥ q (B, G) follows immediately from (5). For the proof of (5) we will need the following definition and a lemma.…”

On the second weight of generalized Reed-Muller codes

“…The same method was used in [5] to deduce the minimum distance of the improved generalized Reed-Muller codes known as hyperbolic codes or Massey-CostelloJustesen codes. The original method used to derive the minimum distance of the generalized Reed-Muller codes ([6, Th.…”

On the evaluation codes given by simple $$\delta $$ δ -sequences

“…It is not hard to show that the improved dual codeC(l), 0 < l ≤ q 2 , for the δ-sequence G coincides with C φ (l). In addition, reasoning as in [23], the equality of primary and dual improved evaluation codes given by G can be proved (see also [5]) and also that l is the actual distance ofẼ(l).…”