In this paper, we introduce a lower bound for the generalized Hamming weights, which is applicable to arbitrary linear code, in terms of the notion of well-behaving. We also show that any [n,k] linear code C over a finite field F is the t-th rank MDS for t such that g(C)+1 t k where g(C) is easily calculated from the basis of Fn so chosen that whose first n-k elements generate C. Finally, we apply our result to Reed-Solomon, Reed-Muller and algebraic geometry codes on Cab, and determine g(C) for each code.

Higher transmission rates are one of the main characteristics of the fourth-generation (4G*) of mobile communications. These systems are expected to operate at higher frequency bands, which experience larger propagation loss. This results in larger required transmission power, which causes several problems, particularly for uplink communications, as the typical mobile station (MS) has limited transmission power. Multi-hop systems have been proposed to address this problem. In this paper, we consider the issue of random-access (RA) in a multi-hop system. It is clear that a two-hop mobile communication system requires a two-stage RA process. In this paper, we propose a two-stage RA process that is an extension of the RA process of the CDMA-based 3GPP standard. The proposed method uses a hybrid of code division multiple access (CDMA) and Slotted-ALOHA. To realize the proposed two-hop RA, we dedicate one slot for second-hop transmissions in each interval (predefined); we refer to this as the interval slots allocation (ISsA) technique. Numerical analyses and simulations are conducted to evaluate its basic performance in a multi-hop system. The results demonstrate the superior throughput-delay performance of the proposed two-stage RA multi-hop system with ISsA.