Rubber composition based on a reinforcing filler and on an elastomeric matrix comprising a highly saturated diene elastomer, characterized in that the highly saturated elastomer comprises, distributed randomly within the elastomer, - units of the following categories A, B, C, D and E, or - units of these same categories A, B, C, D and E comprising, for at least one of the categories B, C, D and E, a mixture of at least two different units of the same category which differ from one another through the nature of their side groups Ri: A) -CH 2 -CH 2 - at a molar fraction of m% B) at a molar fraction of n% C) at a molar fraction of o% D) at a molar fraction of p% E) at a molar fraction of q% where R1, R2, R3 and R4, which are identical or different, denote a hydrogen atom, a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, R5 denotes a hydrogen atom or a methyl radical, R6 denotes a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, m, n, o, p and q are numbers ranging from 0 to 100; and in that the highly saturated elastomer has the following properties: • m / p ≥ 1.75 • m ≥ 20 • n + o ≥ 10 • o + p ≤ 30 - units of the following categories A, B, C, D and E, or - units of these same categories A, B, C, D and E comprising, for at least one of the categories B, C, D and E, a mixture of at least two different units of the same category which differ from one another through the nature of their side groups Ri: A) -CH 2 -CH 2 - at a molar fraction of m% B) at a molar fraction of n% C) at a molar fraction of o% D) at a molar fraction of p% E) at a molar fraction of q% where R1, R2, R3 and R4, which are identical or different, denote a hydrogen atom, a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, R5 denotes a hydrogen atom or a methyl radical, R6 denotes a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, m, n, o, p and q are numbers ranging from 0 to 100; A) -CH 2 -CH 2 - at a molar fraction of m% B) at a molar fraction of n% C) at a molar fraction of o% D) at a molar fraction of p% E) at a molar fraction of q% where R1, R2, R3 and R4, which are identical or different, denote a hydrogen atom, a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, R5 denotes a hydrogen atom or a methyl radical, R6 denotes a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, m, n, o, p and q are numbers ranging from 0 to 100; R1, R2, R3 and R4, which are identical or different, denote a hydrogen atom, a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, R5 denotes a hydrogen atom or a methyl radical, R6 denotes a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical, m, n, o, p and q are numbers ranging from 0 to 100; • m / p ≥ 1.75 • m ≥ 20 • n + o ≥ 10 • o + p ≤ 30