The phylum Nucleocytoviricota consists of large and giant viruses that range in genome size from about 100 kilobases (kb) to more than 2.5 megabases. Here, using metagenome mining followed by extensive phylogenomic analysis and protein structure comparison, we delineate a distinct group of viruses with double-stranded (ds) DNA genomes in the range of 35-45 kb that appear to be related to the Nucleocytoviricota. In phylogenetic trees of the conserved double jelly-roll major capsid proteins (MCP) and DNA packaging ATPases, these viruses do not show affinity to any particular branch of the Nucleocytoviricota and accordingly would comprise a class which we propose to name "Mriyaviricetes" (after Ukrainian Mriya, dream). Structural comparison of the MCP suggests that, among the extant virus lineages, mriyaviruses are the closest one to the ancestor of the Nucleocytoviricota. In the phylogenetic trees, mriyaviruses split into two well-separated branches, the family Yaraviridae and proposed new family "Gamadviridae". The previously characterized members of these families, Yaravirus and Pleurochrysis sp. endemic viruses, infect amoeba and haptophytes, respectively. The genomes of the rest of the mriyaviruses were assembled from metagenomes from diverse environments, suggesting that mriyaviruses infect various unicellular eukaryotes. Mriyaviruses lack DNA polymerase, which is encoded by all other members of the Nucleocytoviricota, and RNA polymerase subunits encoded by all cytoplasmic viruses among the Nucleocytoviricota, suggesting that they replicate in the host cell nuclei. All mriyaviruses encode a HUH superfamily endonuclease that is likely to be essential for the initiation of virus DNA replication via the rolling circle mechanism.