In today's digital environment, large volume of digital data is created daily and this data accumulates to unforeseen levels. Industries are finding it increasingly difficult to store data in an effective and trustworthy manner. Distributed storage appears to be the greatest approach for meeting current data storage demands at the moment. Furthermore, due to disc crashes or failures, efficient data recovery is becoming an issue. At present, new data storage techniques are required in order to restore data effectively even if some discs or servers are crashed. Hence, this proposed work aims to improve the storage efficiency and reliability in distributed storage systems (DSSs) to withstand disk failures and data erasures by modifying the degree distribution of Luby Transform (LT) codes. Recently, deriving an optimal degree distribution in LT codes becomes an upcoming research field. Hence, in this work, a novel approach called Lower Triangular Matrix (LTM) based degree distribution is presented for LT encoding. Unlike the traditional schemes such as Robust Soliton Distribution (RSD), there is no need for transmitting the neighbor symbols information along with each and every encoded symbol using LTM-LT codes which results in an efficient utilization of the bandwidth. Further, the overhead of LTM encoder is reduced by performing only one XOR operation for the generation of every new encoded symbol. The performance of LTM-LT codes is evaluated in terms of varying data erasure and disk failure conditions. Simulation results show that the proposed LTM-LT codes perform better compared to the traditional schemes such as RSD used for LT codes in Distributed Storage System.