Highly nonlinear balanced Boolean functions are of the essential primary cryptographic primitives in symmetric design. The Walsh transform serves as a crucial tool for understanding the cryptographic characteristics of Boolean functions. This paper introduces a method for constructing a class of Boolean functions with at most 2k values in their Walsh spectrum and exhibiting high nonlinearity, using the Maiorana-McFarland bent function. The cryptographic properties and distribution of Walsh coefficients are thoroughly examined for this category of Boolean functions, specifically focusing on the case where \(k = 4\) . Additionally, a subclass is developed within the aforementioned set of Boolean functions, where the Hamming weight of each member equals its nonlinearity, which is achieved through the imposition of certain constraints during the construction. Boolean functions serve as essential building blocks for constructing binary linear codes, with Reed-Muller codes and Kerdock codes emerging as prominent examples within this domain. This method generates a class of functions which can be used for construction of binary linear codes.