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In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length 2 ν + δ , size M , minimum subspace distance 2 j based on m -dimensional totally singular subspace in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over finite fields 𝔽 q of characteristic 2, denoted by (2 ν + δ , M , 2 j , m) q , are presented, where ν is a positive integer, δ = 0 , 1 , 2 , 0 ≤ m ≤ ν , 0 ≤ j ≤ m. Then, we prove that (2 ν + δ , M , 2 j , m) q codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ℳ (m , 0 , 0 ; 2 ν + δ) , where ℳ (m , 0 , 0 ; 2 ν + δ) denotes the collection of all the m -dimensional totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code (2 ν + δ , M , d) q in ℳ (2 ν + δ) are provided, where ℳ (2 ν + δ) denotes the collection of all the totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. [ABSTRACT FROM AUTHOR]

Subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang–Xing–Safavi-Naini bound, anticode bound, Ahlswede–Aydinian bound and Gilbert–Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erdős–Ko–Rado theorem. [ABSTRACT FROM AUTHOR]