# Binomial Number System
## 4 Generation of Constant-Weight (CW) Binomial Code  
### Binomial Counting Algorithm  
BNS, which generate binomial numbers and codes, can potentially be helpful for algorithms that solve practical tasks in information coding and the construction of reliable and high-speed digital technology. In conceptual terms, some of the possible applications of binomial numbers are implemented either in programs or schematic form for specialized digital devices and systems, data compression, and error detection codes. However, the practical application of binomial numbers does not end with this list. Almost everything practically implemented based on binary numbers can be implemented on binomial numbers, but in some exceptional cases with greater efficiency. These numbers' efficiency use on binomials is the same as the structure of combinatorial objects built on combinations. This makes it easy to transform from these objects to binomials and vice versa. As a result, it is possible to propose a unified method for constructing combinatorial objects based on binomial numbers. In this case, there is no need for each type of combinatorial object to develop its process of creating them. The binomial counting algorithm consists of two steps, the transition from several combinatorial objects to the binomial number. Then, a change is made, respectively, either to a combinatorial object or to a number. In the first case, the task of constructing combinatorial objects based on code combinations is solved, and in the second, their numbering. A universal approach is used for solving tasks of creating and numbering combinatorial objects, which makes it possible to simplify them and improve the quality of solutions. We developed the binomial counting algorithm based on binomial numbers to solve many practical coding tasks independently and as part of binary digital devices and systems. The binomial counting algorithm consists of the following steps:  
1. An initial code combination consisting of \(n-k\) zeros is formed. 2. The LSB of the code combination is set to 1, which is assigned a zero to the right. 3. Step 2 is repeated until the number of ones in the code combination becomes equal to \(k-1\). After that, one is set into the LSB, but 0 is not assigned. 4. The LSB containing zero is set 1, and zeros are assigned to the right so that their number in the number becomes \(n-k\). 5. The number of units in the code combination is determined. If it is equal to \(k\) and the ones not contained in the \(k\) MSB, the algorithm returns to step 2. 6. If \(k\) ones are contained in the code combination of \(k\) MSB, the algorithm stops. 7. If the number of ones in the code combination is not equal to \(k\), then to the right of the LSB containing 1, zeros are set until their total number becomes \(n-k\). 8. Return to step 2. This algorithm makes it possible to obtain all binomial codewords similar to the codewords shown in Table 1, but with a considerable code length, containing tens or more digits, which is a determining factor for many practical applications. The considered algorithm can be used for a software implementation or microprocessor system (MPS). However, due to the uneven length of binomial codes, it is practically difficult to implement it in the form of a digital circuit with rigid logic, for example, on a field-programmable gate array (FPGA). Therefore, for particular tasks, there are more complex binomial calculating algorithms than the one presented above, using uniform binomial numbers, with the help of which several specific binomial digital devices and systems are constructed [20,21]. Uniform binomial codes, as indicated above, can also be obtained by replacing non-uniform binomial codes with uniform ones by equalizing the \(r\) of the first zeros, as shown in Table 2. As a result, they acquire the property of error detection, which can be practically used to detect errors due to the natural redundancy introduced into the structure of these devices at the time of their construction.