## 1 Introduction  
Low Density Parity Check (LDPC) codes, belonging to the class of block channel codes, were introduced for the first time in the 1960s in [PERSON]'s doctoral thesis [1]. Now they are counted among the most promising and cutting-edge coding techniques in contemporary channel coding. Due to the technological impediments of the time in which they were introduced and of the decoder complexity, these codes were hardly contemplated for about 30 years, with the exception of [PERSON]'s graphical description, introduced in [2], and successively designated as the Tanner graph. Appearing independently of [PERSON]'s work, the authors of [34] were the re-inventors of LDPC codes in the mid 1990s. Afterwards, because of their astonishing performance approaching the Shannon limit along with Turbo codes [5], they were rapidly comprehended in current communication standards [6; 7; 8]. The asymptotic performance of a block channel code, such as an LDPC code, is determined by its minimum distance, i.e., by its minimum weight codewords and by their multiplicities. These determine its correction capability and its asymptotic performance expressed in terms of bit error rate (BER) and frame error rate (FER) vs. the signal-to-noise ratio (SNR) \(E_{b}/N_{0}\) from moderate to high SNRs, i.e., away from capacity. Moreover, as far as LDPC codes are concerned, as first observed in [1], it presents the well-known \"threshold phenomenon\". In other words, the noise threshold defines a channel noise upper bound under which the lost information probability can be maintained as low as needed. Since it is extremely difficult to study the performance of a LDPC codes ensemble analytically as the determination of its weight enumerating function is a complicated task, in this work we try to give an insight into this problem analyzing the importance of the product \(\lambda^{\prime}(0)\rho^{\prime}(1)\) in influencing not only this function, that, as said above, determines the asymptotic BER and FER performance, away from capacity, but also the decoding thresholds, i.e., the performance next to the Shannon limit. Namely, on the basis of the results of [9; 10], respectively, we are interested in determining, on one side, the relationship between the above mentioned product and the minimum distance of a LDPC codes ensemble, and, on the other side, the relationship between this product and the noise threshold of a LDPC codes ensemble. The first, namely the minimum distance, is related to LDPC's codewords weight distribution, i.e., to their distance spectrum. The second, namely the noise threshold, is related to the convergence behavior of their iterative decoding algorithm, which is particularly important because LDPC codes, as well as turbo-like codes, are \"capacity-achieving\" codes. As far as the first of the above cited papers, namely [9], is concerned, its main result asserts that the minimum distance growth rate of irregular LDPC codes ensembles is only determined by the value of the product \(\lambda^{\prime}(0)\rho^{\prime}(1)\), a number that is related to the degree distribution polynomial couple \(\lambda(x)\) and \(\rho(x)\). In particular, if \(\lambda^{\prime}(0)\rho^{\prime}(1)>1\), the minimum distance growth with the block length being sublinear, if not, the minimum distance growth with the block length is linear. As far as the second of the above cited papers, namely [PERSON] et al. 's 2018 paper [10], is concerned, in this work we derived three computationally light upper bounds to LDPC belief-propagation decoding thresholds, usually determined \"exactly\" through density evolution, assuming, as in [11], the validity of the Gaussian Approximation (GA) (the employment of which is very convenient when an effective and computationally light method is required.