# Design and Application of Secret Codes for Learning Medical Data
## 1 Introduction  
With the fourth Industrial Revolution, the application of artificial intelligence technology is expanding in the medical field [1, 2, 3, 4, 5, 6]. The biggest obstacle to the collaboration of medical data from distinct institutes has been the protection the private information contained in the distributed system. In particular, federate learning is in the spotlight as a distributed machine learning technique that can simultaneously retain privacy and efficiency [7]. It can produce a similar result to learning all data at once without sharing private data. Since federated learning does not centralize data to a big server, it can protect the private information of each user. Currently, it is being applied to several areas, including health care, smart factories, and finance [8, 9, 10, 11, 12, 13]. Major companies such as Google and NVIDIA have been conducting research on medical artificial intelligence through the development of their own federated learning algorithms [8, 9, 10].  
Linear block codes have been investigated for applications in several areas of engineering, such as communication systems, cryptography, and security [14]. A minimal code is a block code in which the support of a codeword is not included in that of any other codewords [15]. Using the minimal code, one user's information is not subordinate to other users' information. This has been constantly studied as one of the mathematical structures that can be used in secret-sharing schemes [16, 17, 18, 19, 20, 21, 22, 23]. Furthermore, a minimal code can be used in federated learning due to its distributed characteristics of secret information. Almost all the known minimal codes so far have been designed based on the structure and characteristics of finite fields [24]. In particular, for binary cases, several design methods have been proposed [16, 17, 18, 19, 20]. On the other hand, non-binary minimal code has been investigated recently [21, 22, 23]. Research on previously known minimal codes has focused only on the weight distribution of the codes. Considering recent applications, further characteristics or structures of minimal codes, such as their error-correction capability and relation to the learning rate, should be investigated.  
In this paper, we propose a framework for secret codes in application to distributed systems. Then, we provide new methods to construct such codes using the synthesis ordecomposition of previously known minimal codes. The numerical results show that new constructions can generate codes with more flexible parameters than original constructions in the sense of the number of possible weights and the range of weights. The secret codes from new constructions may be applied to more general situations or environments in distributed systems.  
The remainder of this paper is organized as follows. In Section 2, we present some preliminaries on secret-sharing schemes, finite fields, and minimal codes. We present design methods for secret codes, and propose a framework for the application of designed codes to distributed learning systems in Section 3. We provide the results of our construction in Section 4, and then compare them with the previous researches in Section 5. Finally, we present concluding remarks in Section 6.